Have you thought to try this?
Wet & Forget | The Ultimate Mold & Mildew Remover & Shower Cleaner.
I know it works wonders on moss in the PNW. But it is sloooow. Six months of ignoring it. One application to my driveway did take the oil stains away, though.
Really don't know how well it works on decks, I haven't had an opportunity to try it yet.