TITLE: Question about Pick's Theorem
QUESTION [12 upvotes]: Is there a Pick's Theorem for a general lattice in $\mathbb{R}^{2}$?

REPLY [15 votes]: Certainly there is.  One such version is stated as Theorem 4.1 in this paper of mine:
Theorem (Pick's Theorem): Let $\Lambda$ be a two-dimensional lattice in $\mathbb{R}^k$ with 2-volume $\delta$.  Let $P$ be a $\Lambda$-lattice polygon containing $h$ interior lattice points and $b$ boundary lattice points.  Then the area $A(P)$ of $P$ is equal to $\delta \cdot (h + \frac{b}{2} - 1)$.  
As a reference to the proof I give a 2003 book of Erdős and Surányi.  But -- especially for the $k = 2$ case that you asked about -- Jim Conant's comment is right on: the proof consists simply of making a linear change of variables to get from $\Lambda$ to $\mathbb{Z}^2$.