There aren't too many flat rides in the Asheville area—the Asheville to Marshall River Ride is a notable exception. Setting out from the Asheville Botanical Gardens, riders can take Riverside Drive all the way to Marshall along the river. This is about a 39-mile ride round trip. While the scenery is very nice in most places, the motorists aren't always as nice, so this ride is best done on the weekend if you want to avoid the worst of the traffic.
The Elk Mountain Scenic Highway is far from flat, but the 60-mile cycling route is very popular for Asheville roadies looking for some beautiful views, along with a challenge.
The 51-mile loop around Lake Lure starts out with the some easy hills, gets to some lung-buster climbs midway around, and finishes with some steep drops. The total elevation gain is 4,114 feet. Start at Hickory Nut Gap Visitors Center, then circle the lake on the right hand side up to Old Fort, then come back down through Bat Cave. This is a seriously beautiful location and a must-do ride.