There's a couple of miles of trails in the back of Ben Burton park along the Oconee River. If you look around you can find remnants of an old bridge and some other cool stuff.
From Singletracks: "Take the tallassee road exit on loop 10. Turn away from athens (west). take first left, mitchell bridge road. Ben Burton park is down on your right."
http://www.tripleblaze.com/trail.php?c=3&i=2259