This may have been mentioned and I just missed it, but during times of heavy rainfall I temporarily cover my working compost pile. My advice is to use whatever decomposables you have handy, and then try to control the amount of moisture your pile gets at any given time. If it's already too soggy just spread it into smaller piles until it drains and dries a bit. This will slow the composting process, but it will pick right back up once it's re-stacked.
Once you get a pile up and going and you're using from it, you'll see that the process is easier than it first appears. Here's mine now. It's early spring here, so I've used from the center (most finished) area to begin feeding my plants. The center isn't filled back in yet because it isn't necessary; we're still just a bit too cold for the composting process to begin again yet this season. Having the center open while using from it lets any early rain (or snow-melt) drain away very well. Once I've used all that's ready, I'll re-stack what's left over. By that time the heavy spring rains will no longer be of too much concern, but if we do get a long spell of it, I'll cover the pile until it quits.
