  Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be
its edge ideal. Under a natural condition that the underlying (undirected)
graph of $\mathcal{D}$ contains a perfect matching consisting of leaves, we
provide several equivalent conditions for the Cohen-Macaulayness of
$I(\mathcal{D})$. We also completely characterize the Cohen-Macaulayness of
$I(\mathcal{D})$ when the underlying graph of $\mathcal{D}$ is a bipartite
graph. When $I(\mathcal{D})$ fails to be Cohen-Macaulay, we give an instance
where $I(\mathcal{D})$ is shown to be sequentially Cohen-Macaulay.
