I had to consult my Make It Last Book, by Yankee Magazine, for this answer. Keep polishing! If you continue applying furniture polish on a regular basis, the rings will eventually disappear. Also, to clean a leather tabletop: Mix a solution of equal parts lemon juice and water. Sponge on solution, then wipe dry.
This may not be the quick answer you may have been hoping for but the tips in this book are tried and true Yankee Ingenuity!