To bleach out dark rings on wood surfaces, completely re-move the old finish with a chemical stripper. Place the piece outdoors in the sun and let sunlight start the bleaching action. Swab laundry chlorine bleach onto the wood at full strength. Keep the top surface wet by repeating the applications every 10 minutes until the stain disappears. Rinse with clear water, leave the piece in the sun to dry, stain and refinish.

Source: Decorating magazine