One piece of advice I've seen is to use bait to lure the snails out, and then physically remove them. This may help you keep up with them.
Methods of baiting range from simple (put a large piece of lettuce in your tank, turn down the lights, and wait a few hours) to more involved (build a small "trap" by placing food the snails will find attractive in a container big enough to hold a bunch of the snails, but too small for your fish to get into to eat the bait).
Another popular non-chemical option is to find fish that eat snails, and add them to your tank. Loaches appear to be good choices, if available; especially the clown loach or yoyo loach.
Puffers are also mentioned, but they can be aggressive towards other fish, so probably aren't a good solution.
You can also help keep their numbers in check by cutting down on their available food. Avoid overfeeding your fish, and periodically clean your gravel. You can also add algae eating fish, such as sucker fish (be careful which kind of suckers you get, as some can grow quite large, and therefore require a larger aquarium to ensure their health).
I found some good overall resources/guides for snail control: