Grass clippings work. I tried them one year.
I was a bit alarmed in spring when I saw some (unknown) animal had been digging or flicking the mulch around. I put it back, and when I looked out the next morning saw the culprit - a raven. I was afraid they were after the garlic but not so. When I took another look, I realized the garlic was trying to come up through the thick matted clippings, and bird was actually doing me a favor. Must've been looking for worms.
I used grass clippings every year I grew garlic at my mom's, because she always has a pile of clippings. That worked just fine.