We study excitonic energy transfer in a network of carbon nanotubes (CNTs), a promising light-absorbing material for next-generation organic solar cells. We calculate the exciton energy dispersion curves through solving the Bethe-Salpeter equation in the basis of tight-binding wave functions. Furthermore, we compute the Coulomb-coupling matrix element between bright excitonic states, in order to obtain the exciton transfer rate between similar and dissimilar carbon nanotubes with parallel or perpendicular orientations. The conservation of momentum imposes a limitation on the energy transfer rate between parallel nanotubes of different chiralities. However, there is no such limitation for transfer between misoriented CNTs, which results in transfer rates of the same order of magnitude between carbon nanotubes of similar and dissimilar chiralities. In addition, it is possible to increase the transfer rate by taking the advantage of exciton thermalization and high density of states at the bottom of excitonic subbands.