That is what I use as nesting material as well as floor bedding. I sometimes purchase alfalfa hay if Timothy is not available at Farm store. I prefer hay to straw, because hay is eatable. while straw is not. (straw has about zero nutrition, so is not eatable in that sense)
Unrelated to nesting, but did want to somewhat clarify how I use grass clippings in my run.
Many here are aware of this, but there are always new readers that could benefit from this clarification. I always spread out the clippings as I drop them. A 3 to 4 inch thick/deep layer is what I usually rake to. If you would leave in a pile,,,,,,,,,,, in no time, (like next day) the pile starts to decompose, and turn hot. Then somewhat unpleasant odors accompany the decomposition. When spread out thin layer, the decomposition does not start, and the grass dries into really nice hay type ground-cover.