Most countertops have tile backsplashes that go up to the bottom of the cabinets.

As far as having no backsplash whatsoever, you should be okay so long as there's a bead of caulk sealing the countertop and wall so moisture can't get behind the lower cabinets and grow mold. You'll also have grease on the walls, but that can be washed off with a good commercial degreaser.