There are no dedicated mountain biking trails in Great Smoky Mountain National Park [1]. While you can bike on any park road, many are unsuitable because they are narrow, steep, and clogged with cars. The best biking experience is the 11-mile Cades Cove Loop [2], which is closed to vehicular traffic until 10 a.m. on Wednesday and Saturday mornings from May to September.
You can rent bikes ($4) during the summer and fall from the Cades Cove store (865/448-9034), which is open daily 9 a.m.–5 p.m., with a 7 a.m. opening time on Wednesday and Saturday during the biking season.
Bikes are not allowed on park trails, with the exception of the Gatlinburg Trail [3], Oconaluftee River Trail, and the Deep Creek Trail. Less-traveled park roads that are usually good for biking are Greenbrier [4] and Tremont Roads.
In the state of Tennessee, all children 16 years and younger are required to wear a helmet when they ride a bike. Adults are encouraged to do so as well.
Links:
[1] http://www.moon.com/destinations/tennessee/great-smoky-mountains
[2] http://www.moon.com/destinations/tennessee/great-smoky-mountains/exploring-the-park/smoky-mountain-driving-tours/cades-cove-loop
[3] http://www.moon.com/destinations/tennessee/great-smoky-mountains/recreation/hiking/day-hikes/casual-strolls
[4] http://www.moon.com/destinations/tennessee/great-smoky-mountains/sights/greenbrier