We study directed network design problems with orientation constraints. An orientation constraint on a pair of nodes u and v states that a feasible solution may include at most one of the arcs (u,v) and (v,u). Such constraints arise naturally in many network design problems, since link or edge resources such as fibre can be used to support traffic in one of two possible directions only. Our first result is that the directed network design problem with orientation constraints can be solved in polynomial time in the case where the requirement function f is intersecting supermodular. (The case where there no orientation constraints follows from work of Frank [6].) The second main result of the paper is a 4-approximation algorithm for the minimum cost strongly connected subgraph problem with orientation constraints. Our algorithm shows that the problem of enforcing orientation constraints can be reduced to the minimum cost 2-edge connected subgraph problem on undirected graphs. Finally, we study the problem for general crossing supermodular functions and show the following bi-criteria approximation result. Let k denote the maximum requirement of any set under the given requirement function f. We give 2k-approximation algorithm to construct a solution that satisfies a slightly weaker requirement function, namely, f'(S) = max{f(S) - 1,0}.