Unfortunately, treated hay is becoming more commonplace. But I have to admit (sadly) that the worst lot of hay I've ever used was an "organic" hay - that was full of ragweed, creeping Jenny, and crab grass seed.

Years later, I'm still dealing with the ramifications of that. Here, "marsh hay" is chemically safe, but needs to be inspected for perennial weeds like thistle and yellow dock. Mixed hay, especially grass/clover, seems to be the safest... and the clover breaks down relatively quickly, releasing some nutrients.
One of the benefits of hay vs. straw is that hay feeds the earthworm population, which helps the garden throughout the growing season.
Silly as it may seem (and to the consternation of my lawn-pride neighbor) I intentionally allow my lawn to get a little longer in early Autumn. That way, when I mow & collect the fallen leaves to add to the garden, there is some green mixed in to aid in decomposition. The wood ashes I spread over the leaves probably help speed the decomposition too, as well as adding nutrients to the soil.