In this paper the coupling between two parallel dielectric rod waveguides is investigated analytically. An intergal equation approach is employed in conjuction with Green~ function theory of dielectric waveguides. Cylindrical vector wave functions are used to expand the guided waves inside and outside of the dielectric rod wav~guides. The case of dissimilar guides is examined in detail. Furthermore several coupling geometries are investigated and simplifications are introduced to develop a design procedure for millimeter wave directional couplers.