If you open up a dryer, one surprising thing is the lack of any gears on the tumbler. It turns out that the tumbler is one giant gear (or actually a pulley), and the motor drives a tiny pulley. Because of the ratio between the huge tumbler diameter and the tiny motor pulley diameter, no other gears are needed!
You can see the thin belt wrapped around the tumbler in the picture above.
A motor drives the small silver pulley that is visible beneath the black pulley in the picture above. The belt loops through the silver pulley, through the black pulley and then around the tumbler. The black pulley provides tension -- when the belt is hooked up, the pulley is drawn off center, and the spring tries to pull it back. This gives the belt its tension.
The same electric motor drives both the fan and the tumbler. The pulley for the tumbler belt is hooked up to one output of the motor, and the fan is hooked to the other.