Molecular crystals are attractive candidates for solar energy conversion applications due to their strong light-matter interactions, nearly endless structural tunability, and the relative inexpense with which they can be synthesized and processed. In organic semiconductors, an important step in the energy conversion process is the diffusion of a photo-excited exciton to a donor-acceptor interface where charge separation of the strongly-bound electron-hole pair may occur. In this talk, we present a framework, based on ab initio density functional perturbation theory and many-body perturbation theory within the GW plus Bethe-Salpeter equation approach, for computing the rate of exciton diffusion in organic crystals. We apply our approach to select members of the oligoacene family. Through our analysis we build microscopic insight into which lattice vibrations are most important for exciton transport and how the spin state of the exciton affects the diffusion rate.