We report on the anisotropic photodetachment of positronium negative ions, followed by the dissociation into p-wave electrons and positronium atoms, with a linearly polarized laser beam. We have observed a strong recoil effect of the photoelectrons on the translation momentum of the dissociated positronium atoms. With polarization angle-resolved measurements, the asymmetry parameter of the photoemission angular distribution of the ions at a photon energy of 1.165eV was determined to be 1.97±0.04(stat)±0.07(syst), in agreement with a theoretical prediction. The present method can be applied to explore the unrevealed dissociation dynamics of exotic particle systems and their manipulation with polarized light.