So far I've only managed to get an item from battle after using the move Surf. This knocked down a stone pillar that was in the background and gave me a hard stone, I believe.
I have tried a lot of other moves when I see stones, berry trees and sandpiles in the background, but can never get them drop an item. So what all moves will affect these? Also, are they area specific? For example will only certain berry trees be in Route "X"?