This article studies multiple scattering of matter waves by a disordered optical potential in two and in three dimensions. We calculate fundamental transport quantities such as the scattering mean free path $\ell_s$, the Boltzmann transport mean free path $\elltrb$, and the Boltzmann diffusion constant $D_B$, using a diagrammatic Green functions approach. Coherent multiple scattering induces interference corrections known as weak localization which entail a reduced diffusion constant. We derive the corresponding expressions for matter wave transport in an correlated speckle potential and provide the relevant parameter values for a possible experimental study of this coherent transport regime, including the critical crossover to the regime of strong or Anderson localization.