If the wood has a good finish (don't try on bare wood), mix equal parts of baking soda and regular white, non gel toothpaste rubbing the area that was affected by the water.

Lightly dampen corner of a clean, soft white cloth with water and dip into the paste. With circular motion gently buff the marks for a few minutes. Wipe area clean, and buff to a shine. Follow with furniture polish for a great protected shine.

If rings remain after buffing five minutes or so, they may have penetrated the wood; you might have to refinish the piece.

More Ideas

To remove white spots on darker wood, use petroleum jelly, vaseline, and rub over the white water mark.

You can use regular vegetable cooking oil to rub out water marks on real wood furniture. After rubbing with oil, let it sit for fifteen minutes and then polish as you regularly wood.