Lynn from Boussevain, Manitoba has a cultured marble sink that has some dark orange like stains that just won't come out. It may be that they will never come out, but here is what to try.
Grease or Oil stains -- Whiting with Acetone
Food stains -- Hydrogen Peroxide and a few drops of Ammonia
Rust stains -- Liquid rust remover or oxalic acid
In general when working with porous material like marble, you should make a paste of some kind. If your cleaning agent is not already in a paste form, then add your cleaning chemical to some Bicarbonate of Soda to make the paste. Apply the paste to the stain and cover with plastic wrap and tape down the edges to keep the moisture in. Leave it overnight, then rinse off and buff the surface.
More and more it is possible to find specialized marble repair and cleaning supplies at renovation centres. If not, try a marble installer. The Marble Clinic in Tontoro sells DIY Marble Restoration Kits.