Using the language of chip-firing, we can define a theory of divisors on graphs in parallel to the theory of divisors on algebraic curves. We can define the gonality of a graph, like the gonality of a curve, to be the minimum degree of a rank 1 divisor. It turns out that if we know the gonalities of two graphs, then we can find an upper bound on the gonality of the Cartesian product of the two graphs. I'll show many instances where this upper bound is the actual gonality, and also several constructions of graph products with gonality lower than expected. I'll also show that any nontrivial product graph satisfies Baker's gonality conjecture. This is joint work with Ivan Aidun.