When a peg-legged hillbilly chases a girl around the house, pins her in a corner, and proceeds to have anal sex. The "thunder" reference comes from the sound of thunder as his peg leg bounces on the floor during the chase. Appalachian Thunder is often finished off by a solid donkey punch.
"Hell fire, did you hear about Billy Bob running an Appalachian Thunder on his cousin? Sure did, but the jokes on him cause she has AIDS"