No.<br> These functions are called <i>harmonic functions</i>. The simplest example 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 $r\ge 0$.