Grinding uncooked white rice is the vintage method for cleaning granite mortar and pestles, but it isn’t terribly effective at getting them clean. These days, that method is mostly for “seasoning”, preparing a mortar for use by grinding off any leftover bits of unpolished stone so that they don’t get incorporated into any of the food you want to grind.
Since we stopped putting lye in soap, the modern cleaning method is hand-washing with warm water, a mild detergent, and a stiff-bristled brush before towel-drying thoroughly. Granite and marble actually have a relatively low porosity, so washing the soap out with water isn’t as much of a challenge as it would be with highly porous volcanic stones like pumice, but it’s still best to use an unscented soap so there’s no fragrance to be left behind.