Cut along it with a knife, using the rule as a guide to keep the cuts straight. The pieces can now be separated and stained. Glue the veneer onto the surface of the board. Don't worry, just wait for the glue to dry and then sandpaper around the edges until everything's flush. It may take a few days before the board stops feeling sticky.
Many games only require simple round counters.
Smaller counters can be made the same way from dowel. Stain the counters as required to produce different colours and then polish them up. Since the counters are so easy to make, it's worth having at least one spare of each colour. If you want to make a chess set, you're on your own. Dice are very fiddly things to make accurately, but can be sawn from an off-cut of an old plank.