This is a route developed by dual sport motorcycle enthusiast. It's not completely across the country but most of it's on dirt.
http://www.transamtrail.com/This looks like an interesting route just what JeffOYB was asking for I should think that most of the dirt roads would be very doable on a bike no motor needed. Sure you may run into some sand and other loose terrain that would be easier on a moto but anywhere they can go we can too it just takes us a little longer.
I went looking through the site to see if he has a GPS file available, I see he wants $290 for the maps and a rolling cue sheet. I can't see it he also says it can only be done E-W following the roll sheet. Seems that if you had a map and or even better a GPS track you could go either way.
Several years ago I mentioned to Scott that he should write an AZT mountain bike guide book, his answer was no I think it should be out there on line. One so we can make updates and two that it should be free for those who want to make the trip. This site does just that and so much more. Just sayin.
Tim