I was in Asheville and Boone last week for a hub and spoke bike tour. It is certainly possible to ride through the Appalachians, and the weather was great (high temperatures on our rides of around 75, lows during the day around 65 in some spots -- such a contrast to 99 here in south Georgia). The grades also provided immediate cardiovascular exercise which I grew to enjoy!
Jamawani notes the following: "Appalachian culture is highly automotive. A grown man rides a bike only when he's lost his license. And then there are coal trucks." As a counter to this thought, I can report that I saw over 100 other bicycle riders during my week in the NC mountains. However, I will agree with Jamawani that some of the small roads can be dangerous with no shoulder and heavy traffic. I did note that some of the larger roads did have small paved shoulders.
One option for selecting a route through the mountains is to search for local bicycle club cue sheets; if the locals bicycle ride on a given route, then it probably is doable. For example, check the route cue sheets for the Asheville area here:
http://www.blueridgebicycleclub.org/
Route 441 through the Smokies is very scenic, and an early morning ride during a weekday is possible. Some have lived to tell of the ride on the Crazyguy site. I asked park rangers about this ride and here is the reply I received:
Traffic in July is typically lightest on mid-week mornings, however it can still be very heavy. The road is narrow and steep with blind curves and little or no shoulder. Automobile drivers are looking at the scenery, rather than paying attention to the road, so be prepared for motorists doing stupid things, such as passing you while on-coming traffic is approaching in the other lane, or passing you too closely. (I had the extended mirror of a slow-moving pickup truck pulling an RV clip my shoulder once.)
I would not take route 129 south of Great Smokies Park. That is known as Tail of the Dragon and is an extremely popular motorcycle and sport car route. Look on Yourtube for some of the crazy driving on that road.