Cleaning Discolored Marble (Headstone)

We've got our own private "pet semetary" (sorry Stephen King, I couldn't resist) in our backyard, and one of the little marble headstones got placed too close to ground level, and consequently water, dirt and well, dirty water has splashed all over the lower portion of the headstone and discolored it, compared to the upper portion. The stains are not simply on the surface of the marble, but have actually worked themselves into the marble itself. Is there some cleaning solution that anyone is aware of, other than just immersing the stone in water for some period of time, that will lift or lighten these stains?