Good to know about the different kinds of mold, thanks! I went and looked at the wood chips again this morning. They are very pale and feel dry and lightweight, like playground chips. Or like hacked up firewood (the kind you buy from the store so it's dry). I don't have a pile of green chips to do a direct comparison, and I can't say just how dried out they are, but I'd say they're definitely drier than fresh, living wood. I'll rake the chips in the run periodically to get them exposed to the elements, and won't cover the leftover pile, and I'll keep an eye on things.
As for grass, my situation is different. I'm in the 'burbs, so my yard isn't all that big. Also, at the height of the season, I mow every week, or at least every other week, so the clippings are 1-2" long at most. My chickens don't free range and that's the only grass they see, so they LOVED every dumping. I'd dump it in the run and they'd entertain themselves for the rest of the day spreading it around and eating it. By the next morning, the top layer would turn to hay, so then they'd entertain themselves on the second day by flipping the hay over to reveal fresher grass to eat underneath. And so on until it would all dry out and turn to hay. My (good) neighbors across the street, who don't have chickens, developed a serious grass stink problem because they'd collect their clippings in barrels that would sit in the heat for 2 weeks waiting for yard waste pick up, and they'd stiiiiiiink - forget cabbage, that was a human feces kind of stink. So bad that, apparently, somebody called the health department on them! My chickens solved the problem by offering their recycling services, and for the rest of the summer, my neighbors would let me know as soon as they mowed, and I'd dump their clippings in my run as well. So the chickens got clippings quite often, but I never had any problems with mold, everything got spread around and turned to hay very quickly.
So right now underneath the thick layer of chips, there's a whole summer's worth of two backyards' worth of grass clippings, about 5-6 bags of fall leaves, and whatever else I threw in there since the spring. I'll make sure to stir it around periodically.
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My husband wasn't too excited about the chickens at first either, and tried to argue the total number from 5 down to 3 chickens, saying that 5 was too much. Now he realizes that there's absolutely no practical difference between 3 and 5 chickens, and is not only okay with the 5 we have, but wants to add more 
