Tucson sports several good mountain bike trails.
Here are two:
Fantasy Island Trails: 10 miles east of Tucson near intersection of Houghton and Irvington.
50-Year Trail: Drive 20 miles north of Downtown Tucson on Oracle Rd. Turn right at Golder Ranch Rd. The trailhead is a dirt lot just short of a cattle guard. Cross the cattle guard on you bike and pick up the singletrack of the other side of the road and head north.
Good descriptions from Mountain Bike Bill and his website: mountainbikebill.com