I've had very good luck cleaning ours, and our Sunsetter awning on our house and deck umbrellas with Oxi-Clean. It's somewhat less reactive that bleach, but does kill mold and mildew and cleans well for us. I use it on our deck too before a staining. Seems to be pretty good general purpose outdoor cleaner.
Eric (and Mary who is in no way responsible for anything stupid I post)