Adventure Cycling routes can help some. Once you get to Damascus, VA, which is on the TransAm route, there are a couple relatively decent ways to get to Asheville. Illinois has a nice bike map (I assume Julie or someone from AC will pop in with a link), which could take you down to the TransAm. Alternatively, if you drop south in Indiana, the TransAm goes pretty close to Owensboro, KY, which is just across the Ohio.
60-70 miles is going to be a stretch for most tourists. If you're young, in great shape, and pack lightly, it's possible. It's going to be tough to maintain that pace in the mountains from Kentucky down to North Carolina.