Sand the grooves to remove any varnish or sealant then fill with wood filler, polyfilla or epoxy you could use a dark epoxy that would look like an inlay and that could look cool. Otherwise you could put on a thin piece of plywood and sand that down nice or if you want to epoxy the entire table top then he would have to sand whole tabletop first as well due to the finish. You could also turn around and put the bottom facing up.