In monolayer transitional metal dichalcogenides, the two excitonic states which are eigenstates of Coulomb interaction are coherent superposition of electron and holes in the two valleys. The exciton band which couples to the transverse electric mode of light has parabolic dispersion for the center-of-mass momentum, whereas the one which couples to the transverse magnetic mode has both parabolic and linear components. In this paper we show that the signatures of this band structure of excitons can be observed in exciton diffusion experiments. In particular, it is demonstrated that by pumping the system with linearly polarized light the exciton transport is anisotropic compared to circularly polarized pump and that such anisotropy is absent for the cases of two linearly or two quadratically dispersing bands. We show that the results persist for the moderate level of disorder present in realistic systems. Finally, we demonstrate that similar effects can be obtained for positively detuned exciton polaritons, in less stringent experimental requirements compared to the bare exciton case.