Oh, grass clippings are super for the compost! Have you noticed that they heat up just sitting there?
My landlord where I lived in Washington state used to dump grass clippings at the edge of the woods, and I would go down there with a wheelbarrow and haul them back for my compost. Nitrogen gold, that is. But if you just leave it, it also eventually will rot down nicely into something really good. Just takes longer.