What is the best way to clean marble and other stones?

The old rule of thumb is never to use anything you wouldn’t use on your hands.

Never use powdered cleansers or abrasive pads to clean your stone. Even “soft scrub” type cleaners contain pumice, which is powdered volcanic stone, and might damage your stone countertops or floors. Never use any product which is acidic; this includes substances like ammonia or many common liquid cleaners such as Windex. You should always use sealers and cleaning products designed specifically for natural stone.