I recently watched a TV show where a family was having all of their golf balls stolen from their yard. They set up a camera, and it turned out to be a fox. It didn't matter how many golf balls they put out everyday, the fox would come through and take them each night. If you had a fox it would probably be killing your chickens too though.
I also agree with the thought that it might be a packrat. I have had several of these, and they take very odd things. I once had some chicks disappear, and when I found the packrat's nest he had put all of the chicks' heads in one pile, and their bodies in another. It was really gross. To catch a packrat you can use a mouse trap, but use something shinny to lure them in, not cheese like with mice.