On the maximum of a harmonic function on $\mathbb{Z}^d$

Question

Consider a non-constant harmonic function $f$ on $\mathbb{Z}^d$ (meaning this that $f(x)$ if the average of the $2d$ values $f(y)$ such that the distance between $x$ and $y$ is one). Let $M_n$ denote the maximum of the absolute values $|f(x)|$ for all $x$ such that $||x||_{\mathbf{L}^1} \leq n$ (where $n$ is a positive whole number). I am guessing that there is a constant $k>0$ such that, no matter what the whole positive number $n$ might be, $M_n \geq kn$. However, I am not sure how to tackle this. Any suggestions?