Any dry plant matter...straw, hay, dried grass clippings, dry garden refuse.What do you mean by dry organic materials?
Gotta watch the mix so stuff doesn't pack into a wet anaerobic mass...that's where the wood chippings come in, help keep things from packing.
This works for odors, and might loosen the soil underneath, but I wouldn't use it as garden soil because of the wood.
My runs have semi-deep litter(cold composting), never clean anything out, just add smaller dry materials on occasion, add larger wood chippings as needed.
Aged ramial wood chippings are best IMO.