A harmonic function $f:\mathbb R^3\to\mathbb R$ satisfies, for all $\mathbf p\in\mathbb R^3$ and $r>0$, $$f(\mathbf p)=\frac{1}{4\pi r^2}\int_{S(r)} d\sigma\, f,$$ where $S(r)$ is the sphere of radius $r$ with center $\mathbf p$. This is shown for example in these notes (pdf alert).
I suppose a similar result holds for surfaces other than spheres. If this is the case, how is this more general case proved? Both explicit proofs and references to relevant sources are welcome.