The game I'm thinking of was played on a grid of squares (somewhere in the range of 8x8 to 10x10) where each player put down pieces on their side of the board (I think it was white on the bottom rows and black on the top rows). After all the pieces were placed then the pieces would move around automatically with no further input from the player, each type of piece moving according to its own set of rules.
The only piece I can remember had a symbol resembling a reload/recycle symbol (a pair of curved arrows, the head of each pointing at the tail of the other). It's rule was:
If there's a piece of the same color to the right, jump over it. If this would land it on a piece of the same color, jump over that too. If the right end of a row is encountered, jump to the left most square of the row next up.