It is well known that $\mathbb{Z}^d$ has Liouville property, i. e. every bounded harmonic function on this graph is constant.

(harmonic means that the value of $f$ in a point $x$ is equal to the average of $f$ over neighbours of $x$ in the lattice $\mathbb{Z}^d$).

What are the nicest/shortest/most ingenious proofs of this fact that you know?