Transport properties of exciton polaritons in anisotropic quantum microcavities are considered theoretically. Microscopic symmetry of the structure is taken into account by allowing for both the longitudinal-transverse (TE-TM) and anisotropic splitting of polariton states. The splitting is equivalent to an effective magnetic field acting on polariton pseudospin, and polarization conversion in microcavities is shown to be caused by an interplay of exciton-polariton spin precession and elastic scattering. Analytical expressions for the polarization of the emitted light are obtained. In addition, we considered the spin-dependent interference of polaritons leading to weak localization and calculated coherent backscattering intensities in different polarizations. Our findings are in a very good agreement with the recent experimental data.