I believe the only thing that automatically dislodges (or "pops") cobblestone is an explosion. This leaves you with two options:
Use pistons to push TNT into a position where its explosion will only affect the generated cobblestone. This can be done using specifically-placed obsidian to focus the explosion, with a redstone signal sent through the obsidian (as to not destroy the circuitry). The drawbacks of this method include limited resources to craft the TNT (mostly the gunpowder) and the finite number of TNT blocks which must be pre-placed into the mechanism.
Create a system where mobs spawn in a dark room near you, drawing the attention of creepers and utilizing the explosion (as in StrixVaria's comment). This would require obsidian to focus and contain the explosion, plus a way for creepers to see you and trigger. If designed correctly, the creepers could continuously spawn, and their explosions would kill any other spawned mobs - allowing a continuous, unregulated displacement mechanism. The drawbacks of this method lie in the random nature of mob creation and behavior. The explosion could come from a number of angles, and other mobs might not wander close enough to be destroyed by the creepers. It could also take a very long time for creepers, specifically, to spawn in your room.
Your best bet is probably to harvest as much gunpowder as possible, craft TNT, and then load it into a machine which generates cobblestone in several columns around the TNT. That way, you receive a maximum return for each explosion, and don't have to deal with the randomness of creepers or the rules dictating the spawning of mobs. Use water to wash the popped cobblestone into a designated area, and obsidian to protect your redstone circuitry.