Do you need to submit an itinerary to get a Russian visa? And do you have to stick to it?

You have two option to solve this problem:

There are two types of visa you can get: tourist and private (homestay) visa.
The main difference between them is the form of invitation:

1. For the tourist visa you must provide hotel vouchers for the entire time you will be in a Russia. As you are going to hitchhike, it will be very expensive to pay for the hotel you're not going stay at. So I don't recommend that.
2. For the private visa your friend must get and send you the invitation. Officially you will live at your friend's home. There is no regular check for it, just always get the return tickets yourself, and everything will be okay (also see this answer, if you're going to stay more than 3 months).

More information can be found here. For the invitation your friend must contact local FMS in Russia.

I think business visas are a great option, although obtaining one is more costly.

What is important to note, as already suggested by Mark Mayo, is that you are entitled to do tourist travels with your business visa, that is once you obtain it. While consulates demand a Letter of Invitation (LOI) from a Russian company, you do not need to visit this company once you are in Russia or anything like that. You do not need to look like a businessman either ;) Normally, you pay an agency which will organize an LOI for you from a Russian company and you are good to apply. Since the visa is granted by the consul, it is not in the capacity of the border folks to question it, really.

I previously travelled through Russia on a 1-year multi-entry business visa. I crossed to Russia on foot from Abkhazia with a 25-year old backpack and no questions were asked about my business visa.

Another thing are registrations. If I remember correctly, if you stay longer than 3 days in a given place you need to register yourself at that place. This is done by a hotel at which you are staying. What if you are an independent traveller? You can register at anybody's private house at a post office, as long as that person owns the place. Last time, I registered at my friend's place in Cherkessk for 2 or 3 months (sic!) although I stayed there only 1 or 2 days. I got a slip of paper stating my registration. All I had to do later is to claim that I did not stay anywhere else longer than 3 days, which would make me require a new registration invalidating my previous one.

If you need to stay at a hotel, also not a problem. You can show them your private registration that is still valid and tell them not to make a new one. As long as you don't stay long with them, there should be no problem with that. Even if you are far away from the place of your registration, you can say that you are on a trip and that you would be returning to your friend's place soon. But normally no one will ask you for your registration anyway. Still, it's good to have one just in case! When I left Russia no one asked me for it either. I read reports, though, that people were sometimes asked for a registration when leaving the country.

I hitchhiked from the Black Sea to the Caspian Sea and back through all the Caucasian republics, meaning I crossed numerous internal border controls and was even questioned for few minutes by FSB (at the Ingush-Ossetian border). The visa topic was never brought up. So don't worry about travelling as an independent traveller with a business visa in Russia!

On Monday I should receive a new Russian visa, this time 3 months double-entry business visa, and I will be off hitchhiking through Russia again. I will be doing the same, early on I will register somewhere, i.e. at a CSer, for a long period and will not bother again.

Good luck!