Exciton diffusion is crucial for the performance of organic semiconductors in photovoltaic and solid state lighting applications. We propose a first-principles approach that can predict exciton dynamics in organic semiconductors. The method is based on time-dependent density functional theory to describe energy and many-body wave-functions of excitons. Non-adiabatic ab initio molecular dynamics is used to calculate phonon-assisted transition rates between localized exciton states. Using Monte Carlo simulations, we determine exciton diffusion length, lifetime, diffusivity and harvesting efficiency in poly(3-hexylthiophene) (P3HT) polymers at different temperatures, which agree very well with experiments. We find that exciton diffusion is primarily determined by the density of states of low-energy excitons. A widely speculated diffusion mechanism, namely an initial downhill migration followed by thermally activated migration, is confirmed and elucidated by the simulations. Some general guidelines for designing more efficient organic solar cells are obtained from the simulations.