Directed graphs are better than undirected graphs in capturing the connectivity in many applications like the WWW and networks of wireless devices. For undirected graphs, it is well known how to obtain the expected inter-vertex average commute times from the graph Laplacian matrix. We show the same formulas hold in the case of strongly connected directed graphs. Our result is obtained by deriving a close relation between the Laplacian with a particular scaling and the so-called Fundamental matrix for an associated random walk over the graph. We find that the commute times still form a metric and give bounds in terms of the stationary probabilities for the random walk. Using a simple model of wireless devices, we observe that these commute times can differ from those obtained from a previously proposed symmetrized Laplacian derived from a related weighted undirected graph.