Agreed. If I saw someone with a bike on the AT I would probably say something to them and/or call a ranger. It's a horrible idea, especially for some of the busier and more dangerous sections. Aside from that, there are significant portions that would be impossible to complete on any mountain bike.

I'll second what others have said about biking the Blue Ridge Parkway and Skyline Drive. Not only are they beautiful areas, drivers tend to be respectful of bikers. I also think the speed limit for most (all?) of the route is 35 mph.