This theorem is well-known (maybe it can be called Morera's theorem):
A continuous function satisfying the mean value property on balls is harmonic.
I was recently surprised to hear in a talk that the conclusion still holds if you only check the mean value property on three (I think) radii. I also can't remember if this should refer to the mean value taken on the interior or boundary of the balls (if it matters at all). Does anyone have a reference or name for this theorem? I would enjoy seeing the details and a proof.