1. Take the Virginia Creeper trail to Whitetop, U.S. 58 over to Mouth of Wilson, then cut over through Sparta to the Blue Ridge Parkway. Turn right and ride 200 miles or so to Asheville. There's really only 2-3 mountains on the BRP, along with a bunch of little hills. Virtually no services on route except at Boone/Blowing Rock and Little Switzerland, but the grades are 6% or less and the speed limit is 45 mph.
2. Go south through Shady Valley towards Elizabethton. Go down to Erwin (where they hanged the elephant!) and head over the mountain into N.C. I can't really help you with routing into Asheville from there, although with the completion of I-26 a while back, a lot of the traffic may be off U.S. 19.
I know maybe half of the alternate routes through east Tennessee, if you decide to go another way through there.