  A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such
that every path and cycle of length four in $G$ uses at least three different
colors. The star chromatic index of a graph $G$, is the smallest integer $k$
for which $G$ admits a star edge coloring with $k$ colors. In this paper, we
first obtain some upper bounds for the star chromatic index of the Cartesian
product of two graphs. We then determine the exact value of the star chromatic
index of $2$-dimensional grids. We also obtain some upper bounds on the star
chromatic index of the Cartesian product of a path with a cycle,
$d$-dimensional grids, $d$-dimensional hypercubes and $d$-dimensional toroidal
grids, for every positive integer $d$.
