"Out of bounds" and "playground" are the ones I have always heard in the parts of the US where I have lived. "Off limits" has also been commonly used. Playground and schoolyard have had different usages in my experience. A schooyard could also be a playground, if it had playground equipment. A playground could also be separate from any schools. A schooyard with no playground equipment was never, to my recollection, a playground. To be perfectly honest, though, I don't seem to remember either of those being the most commmonly used to refer to the play area outside of schools when I was growing up, but I can't remember what we did use. Must be creeping old age.

But I'm picking up a lot of Newfoundlandisms (and some Canadianisms, too) being married to a Newfoundlaner, so there's justice somewhere.