If it's going to be a really messy, greasy, long cook (such as pork shoulders), I will cover the drip pan/covers with foil. Otherwise, I just wipe what mess I can with some paper towels, or use a metal scraper (you are not going to damage the drip pans; they're heavy 304 stainless).
For the grates/racks, we have one of those grill cleaners with a thick scotch-brite type pad & scraper.
A couple times per year I just stick the grates, and whatever other stainless drip pan/FlameZone parts that fit, into our self cleaning oven for a cleaning cycle.