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.