Rotting wood is supposed to happen in a forest, isn't it? That's what happens when trees and branches fall. I'd break up what's far enough gone, leave the sawdust and small pieces where they are to finish breaking down, then take the larger bits and maybe stack them near that property line to keep the trespassers out but have it still be on your side. If there are any that are still in good sized pieces, you could use them to outline beds for appropriate plantings. Generally, let the natural processes finish it.

Think of it as wood compost. Use to mulch your plants and or mix with your soil to lighten it. Depending on the degree of decomposition you may need to add nitrogen to the soil because the process ties up the usable nitrogen until the process is complete.

If you have some reasonable (not totally punky or rotten) logs and stumps, you can get some mushroom spawn plugs. The ones I've seen available aren't native fungi but it is something you could try if you have wood to do it. There are choice edible and medicinal varieties available. I have also seen morel kits available. They are a native variety that are highly prized. Just an idea. I included a link to give you an idea of what I'm talking about.

FataMorgana

Here is a link that might be useful: Territorial Seed - Mushroom kits & Plugs

Start a compost bin. It will take many years for the wood to rot, but if you reduce the size of the pieces that will help. If you can rent a chipper that will really help speed the process. Alos if you cover the wood with soil it will rot faster.

If it's hardwood, that rotting wood is called "hardwood fines" (I learned that recently!) and is what trilliums and other woodland natives love. Whatever you do, don't get rid of it. Use it for your special woodland plants.
VG

Another benefit of the rotting wood, is that it may attract woodpeckers to your property, as well as other birds and small animals.

Fence posts, trail markers, benches, borders for planting beds and paths and property lines, are a few items.
You could also drill holes in the wood as homes for solitary native bees. The really rotten crumbly stuff can be left as is or used as mulch.

I also have a lot of wood, brush piles, and dead vines everywhere in various stages of decay. Leftover from tree work and the removal of invasive plants. Sometimes I smash it or stomp on it to break it up and decay faster. It's kind of messy but it's in the back, not really obvious to the neighbors, and the wildlife loves messy places.

Someone mentioned creating piles near your property line - you could make wildlife brushpiles with the larger pieces of wood at the bottom, piling up progressively smaller brush, then top with evergreen boughs to make an excellent wildlife habitat. Use brush from sticker bushes - Rosa multiflora, Barberry, etc. if you want to keep people or deer out. I've found that the birds are very attracted to brushpiles.

If you have any larger logs you can "plant" a dead tree - set the log upright into a hole in the ground (like a fencepost) and this provides food and habitat for woodpeckers and other cavity nesters.

I have piles of rotting firewood in the woods, and various creatures like chipmunks, deer mice, Carolina Wrens, etc. use the piles as habitat.