  We consider the numerical approximation of $f({\cal A})b$ where $b\in{\mathbb
R}^{N}$ and $\cal A$ is the sum of Kronecker products, that is ${\cal A}=M_2
\otimes I + I \otimes M_1\in{\mathbb R}^{N\times N}$. Here $f$ is a regular
function such that $f({\cal A})$ is well defined. We derive a computational
strategy that significantly lowers the memory requirements and computational
efforts of the standard approximations, with special emphasis on the
exponential function, for which the new procedure becomes particularly
advantageous. Our findings are illustrated by numerical experiments with
typical functions used in applications.
