Determining the minimum genus of a graph is a fundamental optimisation problem in the study of network design and implementation as it gives a measure of non-planarity of graphs. In this paper, we are concerned with determining the smallest value of $g$ such that a given graph $G$ has an embedding on the orientable surface of genus $g$. In particular, we consider the Cartesian product of graphs since this is a well studied graph operation which is often used for modelling interconnection networks. The $s$-cube $Q_i^{(s)}$ is obtained by taking the repeated Cartesian product of $i$ complete bipartite graphs $K_{s,s}$. We determine the genus of the Cartesian product of the $2r$-cube with the repeated Cartesian product of cycles and of the Cartesian product of the $2r$-cube with the repeated Cartesian product of paths.