These functions are called <i>harmonic functions</i>. One the simplest examples is $f(x,y)=xy$.

More generally, the real part of any holomorphic function $\mathbb C\to \mathbb C$ is a harmonic function $\mathbb C\to \mathbb R$.
<Hr>
<i>Added later:</i><br>
As mentioned by Gerald, harmonic functions are characterized by the property that 
$$\int_0^1f(z+re^{2\pi\theta})d\theta=f(z),\qquad \forall r\ge 0,\quad \forall z\in \mathbb C.$$
I don't know whether that property for $r=1$ implies that property for all $r\ge 0$.
<hr>
<i>Partial answer to the edited question:</i><br><br>
If you require the function to be bounded, then I think that yes, that should force it to be constant.
[Liouville's theorem][1]
states that any bounded holomorphic function $\mathbb C\to \mathbb C$ is constant.
There is also a version of Liouville's theorem for harmonic functions, so yes: the function is constant.

<i>Gap in the argument:</i><br>
▹ why is the function harmonic?


  [1]: http://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)