From my experience of cycling in 10 US states (Colorado, Utah, New Mexico, Arizona, Washington, Oregon, Idaho, North Dakota, Minnesota and Tennessee) and coming from Europe as well, I found that in general:

• Traffic is more relaxed than in Europe and many roads have a very wide shoulder. In general I found the cycling more pleasant than on similarly sized European roads. The problem is that it is often harder to find quiet minor roads in the US. My impression is that overtaking cars will give you more space. It can get a bit hairy when to trucks cross each other right where you are.
• I never felt unsafe or threatened for (perceived) crime-related reasons. I assume there are less safe areas, just as there certainly are in the UK.

Don't over think this. The US is a safe Western country.

NB: when I wrote this answer, the question was less focussed on crime than it is now.

Anywhere rural I've ever cycled¹, I felt safer than in rural England.

Although you mention crime in your question, unless you intend to make some side-money smuggling stuff or people across the Mexican border or similar adventures, the risk of getting seriously hurt in crime is essentially negligible compared to the risk of getting seriously hurt in traffic, so I will focus exclusively on the latter.

I've cycled through the countryside in the USA and the UK. Traffic-wise, I find it much safer in the USA:

• Apart from a handful of urban areas, USA is far less densely populated than England (and even many urban areas are so sprawling that the urban area of Atlanta has a population density not much higher than all of England). In remote areas, you'll find roads where traffic is almost nonexistent.
• In both urban and rural areas, roads are much wider in North America, so there is far more space. No such thing as narrow winding roads with views blocked by hedgerows but cars still driving at 90 km/h.
• Contrary to ptityetis answer, I find it easier to find quiet minor roads in the USA. Vast areas of North America are crisscrossed by a regular grid of tiny roads, which means you can cycle straight for hours if not days without being on a major road at any time. In the UK, there are very few straight roads built after the Roman era and you might spend a lot of time puzzling on the map trying to figure out a route from A to B that avoids dangerous roads yet is not a huge detour. In North America, it may simply be the case of hopping one block North and continuing your way west. That actually applies for many urban areas as well.

¹Netherlands, Germany, Switzerland, Austria, France, UK, Sweden, Finland, Norway, Iceland, USA, Canada. I now live in southern England and no doubt cycling conditions are worst among this list.

In my experience long-distance cycling in the US countryside is very safe. I cycled rural MA, CT, NY, PA, OH, WV, MD, VA, NC and NJ. I felt safe crime-wise when cycling poverty-stricken places in the rust belt. The unsafe feeling starts when you roll into the cities.

There are several things to consider, however.

1. When you are following US (or state) bike routes through "real countryside", they may lead you through almost backyards of farms (e.g. PA bike route S near Lancaster, PA). There are a lot of farm dogs who will bark and follow you for some distance. Be ready for that.
2. If you are going through some real wild spots, you should be ready for all kind of wildlife encounters. Get a bear spray.
3. American countryside is much more diverse than English one. Prepare to encounter all sorts of communities, each with their own unspoken rules (e.g. Amish, Mennonites, Mormons). E.g. don't expect Amish girls on a bike to help you with directions.
4. Rural US residents are - in general - much more about protective about their private property. Do not get off the public thoroughfares, without asking the owner of the adjacent land for permission. Do not set your tent on a seemingly unoccupied piece of land.
5. It is common in rural areas to encounter farming machinery on a road. Sometimes this machinery is not very well maintained. Stay away from it. I almost got hit when a disengaged hitch let go of the trailer with some farm equipment.
6. Some rural roads are not paved, so plan according to the weather and tires of your bike
7. Be prepared that to some of the residents you, as a long journey cyclist, will seem like a "pink elephant". So they will stare at you.
8. Do not count on cell phone coverage for navigation. Have a dedicated GPS unit, printed-out maps and general idea of a road ahead.
9. Do not be a bike Nazi

And, obviously, follow the usual precautions, such as:

1. Travel in a small group, if possible.
2. Wear some kind of (satellite) emergency tracker/beacon
3. Make sure someone else is aware of your itinerary.

Otherwise, most of the people I met were extremely friendly and helpful with my small bike problems.

I think you have no sense of how big and how empty the US is. If the US were as densely populated as England, it would have 2.5 billion people. Unless you are in an urban center, you won't find many people of any sort, let alone criminals.

The quickest coast-to-coast route is Jacksonville to San Diego, almost 4000 km, three times the distance from Land's End to John O' Groats, 200 hours of riding for a strong cyclist, six weeks on the road. Here is what it looks like about half-way. See any potential muggers? At about three-quarters. (This is the shortest route, not the scenic route.)

Don't worry about "smugglers" or "gangs". They exist but are largely preoccupied with smuggling and gang-stuff and won't even notice a limey cyclist as they whiz past at 120kph.

Your big concern will be water. Out west, 100km gaps between sources of fresh water are not at all uncommon, and it would only take a little bit of bad luck (a leaky camelbak shorting out your cell-phone for example) to make dying a very real possibility.

You'll also have to cross two, and possibly three, major mountain ranges. I don't know if there is any route you could take that would not require you to climb to a pass half again as high as the highest peak in England.

As a Brit, I'm used to the idea that the countryside is not at all dangerous, that is from a criminal/violent point of view.

This is true in the U.S., also, with the possible exception of drug dealers/smugglers, particularly in the deserted areas near the Mexican-American border. I'd advise not cycling there. Otherwise, rural areas of the U.S. are quite safe (from a crime standpoint.) Larger cities, however, can be another story, especially in the lower-income residential 'inner city' areas. Almost all small cities and towns will be safe, but some large ones are less so.

Frankly, when biking across rural parts of the U.S., violent crime should probably be the least of your concerns. Depending on exactly where you are, weather, traffic, vast distances between even tiny towns (often 100+ miles, sometimes much more,) harsh terrain, and/or wildlife will be much larger concerns. Most of those concerns (except traffic) apply more to the Western states. Once you're East of the Mississippi River, things aren't quite as desolate and there will at least be small towns somewhat frequently (though still not nearly as close as they are in Western Europe.)

Make sure to read up on the normal climate of the areas you'll be traversing in the season you'll be traversing them and also make sure you have some way to keep up with the weather forecasts for where you'll be. Weather in many parts of the U.S. (especially the parts away from the coasts) can change very quickly. Depending on season and where you are, the temperature may be anything from +50 C to -40 C. When I was in North Dakota a couple of years ago, it was +40 C while I was there and they had a snow storm about a week later. Even where I live in the Southeast, temperature swings of 20-30 C over the course of a day or two are not at all uncommon, especially during the non-summer seasons.

Another safety consideration to keep in mind is the type of road you'll be on. It's important to note that bicycle traffic (and any other form of non-motorized traffic) is completely banned on most Interstate highways. Even on several non-Interstate highways, there can be a significant amount of traffic even in the rural areas and quite high travel speeds (70-85 mph / 110-140 km/h is common in some areas.) Biking on those roads, even if it might be legal in some cases, is definitely not advisable.

Also, in case you haven't spent much time in the U.S., it's worth a reminder that the U.S. is very, very large. The distance from New York City to Los Angeles is a little more than the distance from Lisbon to Moscow. You'll be crossing everything from large mountains to desert to plains to hilly countryside. It typically takes about 3-4 days to cross the U.S. by car at 70+ mph / 110+ km/h. Even by airplane, it's similar to crossing the Atlantic. Crossing it by bicycle is not a small undertaking.

I've been to 45 out of the 50 states, especially in rural areas.

Like Hank Williams, Jr. said:

you only get mugged if you go down town

Country criminals might steal your stuff if you leave it unattended, but they won't assault you or rob you in person.

There's an extremely small chance that you could be accidently shot, like this cyclist in Herndon, VA (suburban, not rural) who was accidently shot by someone loading his gun in his house.

(The shooter, John Alders, had to pay a $250 dollar fine for "reckless handling of a firearm".)

Three cross-country routes are:

TransAmerica Trail (try to go through Yellowstone June-September, as cold as -50 degrees C in winter)

Northern Tier (likewise, June-September to minimize chances of snow in the Washington state to Minnesota portion).

Southern Tier

While I will join the crowd in saying the risk of being mugged is trivial there are other threats to be aware of:

Wildlife, including dog packs, is more of an issue in the US than in England. This is very situational, in most areas the threat is about zero but if you're off the main roads and in the areas in the west with evergreen forests the bear threat is small but real. Generally they'll leave you alone but getting between mama and her cubs is a very bad idea.

Also, in the southwest the distances can be long and the weather can be hot. I would recommend reading up on desert survival before biking between cities in the summer in the southwest deserts. Unless you're on the interstate you're unlikely to have cell coverage, an accident on a side road could be a very serious situation. Now and then we have cases of tourists who head out into the desert unprepared, have some sort of breakdown and die out there.

On the flip side, winter storms can be a lot worse than you're used to, also. England is surrounded by ocean and the Gulf Stream warms it to some degree. Much of the US is a lot farther from the moderating effects of the ocean and only a small part of it is warmed by ocean currents.

Finally, our extreme weather events are also worse than anything you're used to. If you're near the ocean in the southeast beware of hurricanes. You're looking at winds of at least 120 kph and the worst of them will be twice this. Rain will be extreme, flooding common. The areas near the ocean are usually subject to a mandatory evacuation in such cases and the areas a bit inland aren't but you'll have problems finding supplies and shelter. In the Midwest you'll also have the threat of tornadoes. They are too unpredictable for evacuation, alerts go out when there is threatening weather, warnings when radar sees a tornado and some communities also have sirens. People take shelter where they can, the well prepared have underground rooms. Winds again start at 120 kph and the record holder was 476 kph. All ordinary houses will be flattened by the strongest of these.

TL;DR: Look, Listen, Live!

While the other answers are indeed correct that crime isn't a major concern (at least compared to weather or even wildlife in some areas), and I don't doubt your ability to deal with traffic hazards to cyclists in general, there are a few peculiar things about North American traffic that could trip you up in a large and splatter-y way.

Trucks are bigger here (just like everything else)

North American standards allow for larger and heavier trucks (lorries for you Brits) than their European counterparts. This is mostly an impact to you when you go to pass a Rocky Mountain doubles rig (this is a 40-50' semitrailer with a 20-28' full trailer behind it) or some other equivalent setup that simply would be too large to fit on European roads. Even larger setups are possible -- up to and including seeing two 40' to 50' trailers in a doubles setup in some parts of the country, or triples trucks with 3 28' trailers one after the other.

There are also long "dump and pup" rigs that consist of a dump truck connected to a full trailer by a long (at least 15') drawbar, which means you need to pay attention for the presence of that drawbar lest you merge back in too early and get clobbered by the pup trailer. Two-pup setups are even seen in some parts of the western U.S., but are not nearly as common as a standard dump-and-pup.

Of course, oversized loads are present on the North American highway system as well, and merit the same cautions as anywhere else in the world.

You may not notice the first few rural grade (level) crossings you go over

Grade crossings (level crossings for the OP) in North America are dramatically different from those in the UK, especially in rural areas and on lesser-traveled roads. In the UK, the railroad right-of-way was required to be enclosed since the early development of the railroad, and controlled signalling systems were implemented over most of the network, meaning that even the most minor crossings are gated and provide a facility to contact rail personnel. Furthermore, most UK rail traffic is scheduled to a published timetable, meaning it is largely predictable in timing.

None of this is true in North American rail practice. The vast distances involved effectively prohibit the fencing of the rail right-of-way save for in urban areas where it's necessary to prohibit trespass, and also made UK-style signaling impractical as well. This means that most crossings in the US are open crossings with no gates, especially on minor roads. (Major roads use a style of crossing that is essentially identical to a UK Automatic Half Barrier or Automatic Barrier Crossing - Local save for lacking monitor functions in most cases.)

Likewise, most of these open crossings lack any active train warning function whatsoever, resulting in what are called passive crossings. These are marked only by the advance warning and crossbuck signs, and require user vigilance to detect the presence of oncoming trains and yield to them. Furthermore, train traffic in North America is largely run on a prioritized "best time" basis, instead of on a regular, published timetable. This means that no matter what time of day or night, you must keep a watchful eye and ear out for rail traffic at grade crossings, because that's the only warning you'll get in most cases, and trains may be present at any time.

(P.S. most North American rail crossings have a contact telephone number on them, but that's an emergency number, not to be used for minor matters such as a cyclist crossing the rail line as part of normal traffic.)

If you stick to public highways and rights of way, the US isn't going to be any more dangerous than the UK in that respect. Where there might be additional risk is if you (intentionally or not) stray onto private property, because the owners are far more likely to be armed.

I'm going to take a slightly different note than all the other answerers unfortunately. While I don't think you'll be targeted for violent crime or theft (assuming you have the good sense to identify bad neighborhoods and stay out of them), it may be worth noting that certain passing motorists may buzz you in non-injurious ways. Examples are -- Intentionally swerving to get close to you (without hitting), honking/yelling as they pass you, or a particularly gross one, revving their engines to cover you in soot/exhaust fumes. This is far from the norm (I can also attest a lot of people will move to the inside lane to give you space, even if you are on the shoulder), but it is something that a certain type of individual (read - Douche) will do, and have done to cyclists just because they're cyclists.

The city is only dangerous because there are more people.

Actually, the chance of any random American hurting you is about the same, city vs. country. (I'm being charitable; actually the per capita injury rate averages a tick higher in the country.) Crime per se is indeed higher in the city (though not as different as it used to be), but there are other risks in the country which you can't avoid, predominantly, traffic accidents.

But that doesn't tell the whole story. We have a lot of localized problems. For instance cities are even safer than statistics say, because their average is pulled down by local problems, e.g. Baltimore and southside Chicago. That same thing happens in the country, but where cities have it almost block-by-block, the country has it town-by-town.

It may be worth calling the state police or better, county sheriffs, along your route, tell them what you're doing and ask about areas to avoid.

As you know, America is broken into the several States, and within each state, 30-120 counties, averaging maybe 50km across.

Especially in rural counties with little gun control, you benefit from criminals having no idea if any particular person has a gun. Don't mess that up by making it obvious. Don't get one, either. The gun laws are a politicized tangle of local and state laws, and accidentally running afoul of one is a huge risk, especially if you're moving across dozens of jurisdictions.

Those other hazards

You will not see an earthquake coming, but, if you're a bicyclist, an earthquake harms your mobility almost not at all. Even if a road is damaged you can simply carry around.

The weather threats, we know they're coming. Watch the news.

Other than that, your biggest enemy is the poor state of our national bicycle-route system. We do have one; it's just nearly zero-funded and is mostly just signage of "State and U.S." highways (both State funded; have level crossings) which have a lot of 55-65 mph (90-105kph) traffic. I'm sorry, did I just explain to a Brit what a mile is? Nevermind me.

America has many pastoral country lanes (often former Fed/state highways), even in the most surprising of places such as frontage roads for rural freeways. Sometimes those are designated or recommended as bicycle routes, but not always: you are often forced onto a very busy main road and sometimes even onto a freeway (M-motorways). Figure on some "on-the-freeway" running unless you choose a cross-America route very carefully.

Your other enemy is that of the Donner Party; the sheer breadth of this country and the raw number of miles you must cross, especially in the west, where services and support can be very far away and cell phone service can be nonexistent. You might want to own a 406MHz EPIRB so you can get help for sure. That said, a lot of people do this.

We have a shocking number of "rails to trails", old railroads converted into bike trails. That's something to think about when choosing routes. They have the advantage of being contiguous and free of motor vehicles. (you had "Beeching Cuts", we had ConRail, but our freight system emerged healthy and profitable.) Often a former eastern rail route will be bike-trail for much of its length.

I'm going to echo the advice of others, you're generally safe from crime on the "random country lanes" you seek, and I won't caution you about your safety from the other vehicles on the road.

Do keep tabs on the weather forecasts, as others have said. It's not just the temperature swings to beware of, there can be downright dangerous weather. Thunderstorms with strong gusty winds, large hail, lightning strikes, and tornadoes are a common thing in the summer months in the middle and eastern parts of the country.

Crime in the US largely falls into violent and non-violent categories. Because of our culture and our media we tend to "report" more one the violent crimes, and just quietly deal with the non-violent crimes.

The largest crime your going to need to be aware of in a rural area is probably going to be theft. Same if your in the city by the way. It's far more likely that your bike is stolen then anything else, but in truth, even that is pretty rare, if you use a U-Lock.

Your two problems are going to be the vastness of the US, an the way our road system works.

The size of it I had a family friend come from England and stay with us in Florida. They decided they wanted to see the country so they decided to drive to California. They were totally unprepared for the shear distance and what it would mean. Our country is large enough that you have different eco systems as you travel, specially east to west. Your will have Swamp, plantations, flatland, prairie lands, deserts, mountain ranges, arid zones and coastal areas (just to name a few). There are areas that get 450 cm of rain a year and others that get 60 mm.

This is also true for people. In some areas strangers are welcome and invited guests. In others they are an annoyance to be avoided. It all depends on area. Now, no one would be hostile. That's not what I mean. But in some areas you will have people invite you over to their house so they can bake you a pie, in others it will be hard to get a passer by to tell you the time of day.

Even animal concerns are going to be different. Most parts of the US, (even in cities at times) have loads of wildlife. Cougars, snakes, (brown or black) bears, alligator, panthers, crocodiles, bob cats, and so on, are very common in rural areas, some even in cities. Smaller animalas are "worse" if your going to be camping out. Raccoons, Possums, Armadillos, squirels, etc. all will have no problem getting into your food stores.

Cell coverage doesn't reach everywhere are there are large stretches of road with nothing on them. Your could easily find your self sleeping rough.

The road system is vastly different in many subtle ways. Most importantly, is the county, state, US highway, Interstate differences. Before the interstate system, if you wanted to travel across country, you would take a US highway or maybe some State Roads. But largely these roads were made to get you from specific point A to specific point B. Small towns and villages popped up along these highways. Then comes the interstate system. This system is designed to take you to maybe 1-2 big cites in a state, but largely to travel long distances between many large locations. They are not meant (primary) for travel inside the same state, though because of their size, they end up being really good at traveling between large cities in the same state.

The result is, if you're in a car, going from one rural area across the country to another rural area, you would take a county road, then a state road, then the interstate. Travel for several days, then get off the interstate near a large city, and take the state and county roads to your rural destination. For a biker, you can't get on the interstate. To do so would be illegal and life threatening. But you can travel the state and county roads quite safely. That said, you, as someone not from the area your traveling in, will have a very hard time figuring out if a road is "safe" to travel. It could be nearly empty, or it could be a main thoroughfare for that part of the state. Even more so because the same road in different areas would be used very differently.