If you have a neighbor with a fireplace, they'll most likely jump at the opportunity to take the logs! If none of your neighbors has a fireplace, you can post flyers on local trees or post in your local freecycle.org chapter, craigslist or other free sites. Decaying logs attract insects, etc. and after a while they don't look very nice anymore
(happened to me
).
It's been my experience that only bundled branches, cut no longer than 4", are picked up when they have regular yard item p/u.