There are more riding options on the North Carolina side of the mountains.

As I mentioned the Blue Ridge Parkway is nice (at least at the southern end). The grades generally don't exceed 5-6%, wide sweeping turns, low traffic (except during peak times) and great scenery.

Other nearby roads can also be nice. For example, here are routes around Sylva, NC.

http://www.motionmakersbikes.com/area_ride_maps.html

There are several nice places to see and/or cycle near Asheville (Mount Mitchell, Lake Lure, etc.).

Roads in the Great Smoky Mountain National Park would be nice, but the traffic is too annoying unless you ride early morning on weekdays.

I understand there's good riding around Boone, NC, but I've never been there (except to Roan Mountain, on the Tennessee side).

The Cherohala Skyway is a challenging adventure. Typically very light traffic except when the motorcyclists invade on the weekends and holidays.

http://www.cherohala.com/