Let C2p and pK2 denote a cycle with 2p edges and p vertex-disjoint edges, respectively. For graphs G, H' and H'', a (H';H'')-multidecomposition of G is a partition of the edge set of G into copies of H' and copies of H'' with at least one copy of H0 and at least one copy of H''. In this paper, we investigate (C2p; pK2)-multidecomposition of the Cartesian product of paths, cycles and complete graphs, for some values p 3.