We introduce a new non-overlapping optimized Schwarz method for anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level, prove its convergence using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions, both for the case of unbounded and bounded domains. We then present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method using energy estimates, and then study its convergence numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.