I've successfully removed oil/grease and even soap stains on my granite kitchen countertops, as well as dried dog urine from unsealed marble flooring (it had soaked in and discolored the floor, as well as saturated the spot with a truly wicked smell). The secret was a poultice, designed specifically for removing grease from stone surfaces. It comes as a powder, which you then mix with water to make a thick paste; after applying it over the stain, you cover the paste with plastic wrap and let slowly dry over 24 hours. You then remove the dried paste with a plastic spatula and wipe clean - the oil gets absorbed into the poultice as it dries (ladies - think clay facial). The poultice I used for the dog pee even recommended putting a teaspoon of household bleach in it, which worked wonders for the stain and smell. I had to apply it twice to really get the smell out, but you'd never know that there was anything there!
Home Depot sells both of the ones I tried: there is a premixed paste that you apply straight from the jar called TileLab Stain Remover; the powder you add water to is Aqua Mix Poultice.