My parents had some trees cut down at their second home down in AR. The stumps were all ground down. The tree guy said to just spread the wood around, to thin it out, so to speak.

Mushrooms typically grow where there are ground stumps. However, I'm in USDA zone 5 -- perhaps you won't get mushrooms where you are. Even so, I just scoop a bit of the grindings away, pack down the rest, so that I can top up with a couple inches of topsoil and then spread grass seed. The grass always grows fine, and I just knock down the mushrooms (which tend to disappear after a couple years. Keep in mind, the grindings decompose so that the spot tends to sink over the next year or two (which is why I pack it down and top it up above ground level). If you have a level pristine lawn, you might want to take away more grindings to ensure it will be level in the future.