  In this short note we prove a sharp dispersive estimate
$\|\mathrm{e}^{\mathrm{i} tH} f\|_\infty < t^{-d/3}\|f\|_1$ for any Cartesian
product $\mathbb{Z}^d\mathop\square G_F$ of the integer lattice and a finite
graph. This includes the infinite ladder, $k$-strips and infinite cylinders,
which can be endowed with certain potentials.
