A furniture renovator friend of mine who renovates pub tables shewed me how to remove ring marks and stains instantly from polished surfaces.
Firstly, using a dry cloth, dampen the cloth with methylated spirit and quickly rub it over the offending marks. Now quickly ignite that area with a lighter or match and watch the marks dissapear in seconds. When the marks are gone, blow out any remaining flame and wipe the top with a clean dry cloth.
Easy!