This question was addressed by Hansen and Nadirashvili in a series of papers, see, for example:
MR1315353 Hansen, W., Nadirashvili, N., On Veech's conjecture for harmonic functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 137–153.
and references there.