I live there
I now live in the former colony and can assure you that it's not in the least bit hard to find the non-touristy places, as this is a vast country that hardly anyone bothers to travel around, preferring to fly over.
Time of year dictates a lot. For instance here in Washington State, there are times of year that you can not reach all the parts of the state, as snowy passes in the Cascades cut off the East from the West. The riding season tends to be three or four months long, unless you do the Nevada, Arizona and California thing and even then higher altitudes will see you caught out by snow (Tioga Pass Yosemite Park for instance).
Then there's the tornado corridor and that season...it all gets a bit extreme, however, I did ride across from Atlanta to Seattle, taking only back ways and missing large towns. Instead of 4k miles it ended up being 10k. 45 days and only 45 minutes of rain.
you can see the map and read about it at:
www.roughguidesintouch.com/lamble
I found everyone hospitable, even overly friendly (I'm an honourary Hells Lover, the black faction of the Hell's Angels...there's no one I know whiter than me!). Lodging is cheap, food not so much so unless you eat the chain restaurant crap..however you do get twice the amount for your money, so if you can eat the same thing twice in a day, ask them to box it.
There are places around the pan handle of Oklahoma, Texas and the Arizona and Nevada areas, where there's absolutely no one about. You can ride for an hour or so and still not have reached the point on the horizon that you saw an hour previous. It's something beyond any experience that you can get in the UK, unless you've run out of petrol and are pushing.
Take a dib into the BMWMOA site and you may find folks who will put you up for the night or provide local ride info.
If you do get as far as Washington State, then if I'm back from the Three Teas Tour, I'll be happy to have you stay and perhaps join me on the Ice Tea ride, summer 2009, following the Ice Age floods that created the scablands.
Cheers.