How can I visit areas with unpaved roads in a rental car?

What do I risk if I ignore the clause and drive on such trails?

Long story short you risk voiding any coverage you have on the vehicle, which will make you responsible for any and all damages that may happen. Also most modern SUVs are not designed to be take off road and some don't even offer 4x4 capabilities, so it is very possible that you may not even be able to get to where you're going.

So how do you get there?

There are 4x4 and ATV tours like this one for Grand Canyon. So if there is anything worth seeing your will have tours and if not you're likely to have places to rent similar vehicles.

How can you visit those areas? Simply take the right turn. There are quite a few unpaved roads which are clearly marked - just look for them.

I've personally been on a few unpaved roads in Death Valley, Capitol Reef National Park and Monument Valley. Use common sense: Sometimes the roads are quite easily accessible, even if they're unpaved. Drive slowly, watch for potholes and if you don't feel comfortable: Turn back!

However: If your contract says that you're not allowed to drive on these kinds of roads and you get stuck or have any other kind of accident I suppose it'll get quite costly. But as they say: No risk, no fun! :)

We've had this problem for a number of years as we have relatives that live along gravel roads in various parts of Alberta. Having looked at all the local rental companies there are basically two variations of the rental contract depending on the company.

The majority of the companies stipulate that the vehicle can only be driven on paved roads, which means we would be in violation of the rental agreements whenever we visited any of the relatives who live along gravel roads. The others stipulate that the vehicle can only be driven on regularly maintained roads, which in the case of Alberta includes the gravel roads. Talking to the various rental companies one said to us that if they restricted the rentals to paved roads only entire towns would be inaccessible by rental car.

Update: Since I wrote my original answer for this we have now found one local Alberta rental company who explicitly allow some of their vehicles to be driven on the gravel roads rather than allowing it via a loophole, but only those in the SUV classes. We came across them quite by chance because they are now the Fox Car Rental franchisee at Edmonton Airport and I talked to them directly to confirm. My basic advice though remains the same, the major rental companies will generally restrict usage to paved roads, but the key thing is to talk to the local rental offices as they may well operate different policies based on where you are going.

Fined 300 EUR for driving on Non Asphalt Roads. In May 2014, we rented a car via Raddisson Hotels from Fastrent, Tallinn, Estonia. As we were about to drive off the lot, the attendent said be sure not to drive off asphalt roads. We did drive on one well maintained national park sand road. Fastrent charged us 300 EUR for driving off asphalt road. A well hidden line in Conditon 6 of 33 on the back of the contract says "Lesse can drive the car only on the asphalt roads."