Determine if the rings are black or white in appearance. White rings are a good sign; these indicate that water has somehow gotten under the finish on the wood but has not yet gotten into the actual wood. Black rings indicate that water has gotten under the finish and soaked into the wood underneath, which causes a "chemical reaction" in the wood that you may never get out.
Plug in and turn on your blow dryer. Use the lowest heat setting and continuously move the air over the ring. Keep the blow dryer at least six inches from the table surface.
Gently buff the area with a soft, dry cloth. If you do not see any improvement, move on to the mayonnaise method.
Measure two tablespoons of mayonnaise into a bowl and mix in two tablespoons of cigarette ash. Wrap a lint-free cloth around two fingers and use it to pick up approximately 1/4 of the mayo/ash mixture.
Rub the mayo mixture into the table using small circles with gentle pressure. You may have to do this several times before you see results.
Leave the mayo mixture on the ring for two hours if Step 2 did not produce results. Check frequently to make sure the mixture is not drying out. If it is drying, simply apply more of the mixture over it to keep it moist.
Gently wipe off the mayo mixture with the cloth and clean the table. Spray lemon oil furniture polish on the table and buff lightly with the cloth.