  In this manuscript, we propose an efficient, practical and easy-to-implement
way to approximate actions of $\varphi$-functions for matrices with
$d$-dimensional Kronecker sum structure in the context of exponential
integrators up to second order. The method is based on a direction splitting of
the involved matrix functions, which lets us exploit the highly efficient level
3 BLAS for the actual computation of the required actions in a $\mu$-mode
fashion. The approach has been successfully tested on two- and
three-dimensional problems with various exponential integrators, resulting in a
consistent speedup with respect to a technique designed to compute actions of
$\varphi$-functions for Kronecker sums.
