Two reasons not to... You'll end up with particles sticking to residue on the grates - steel wool breaks apart a lot. And because steel wool rusts. Combined with stainless steel you're not doing yourself any favors. If you've ever scrubbed a white wall on a tire with steel wool you'd never use it on a cooking surface like a grate. Maybe cleaning a pan you can wash out, but. I don't know many people who wash grill grates with any level of frequency.