In this paper, we design the first linear-time algorithm for computing the prime decomposition of a digraph G with regard to the cartesian product. A remarkable feature of our solution is that it computes the decomposition of G from the decomposition of its underlying undirected graph, for which there exists a linear-time algorithm. First, this allows our algorithm to remain conceptually very simple and in addition, it provides new insight into the connexions between the directed and undirected versions of cartesian product of graphs.