Wet a soft cloth and sprinkle some non-abrasive cleansing powder on it. Rub the powder into the silver marks until they disappear. Bon Ami Cleansing Powder is recommended on the Mikasa website.
Mix baking soda and water together until it forms a thick paste, about the consistency of toothpaste. Dip a damp cloth into the paste and rub it on the silver marks. Let it sit for five minutes, then rinse it off.
Dab silver polish onto the marks if the other methods don't work. "Good Housekeeping's" cleaning expert, Heloise, recommends this if the silver marks are persistent and won't disappear. The silver polish will buff the stains out.