In this work we address Li-ion diffusion in thio-LISICON materials and in their oxide counterparts, exploring both the orthorhombic and tetragonal phases of Li 10 GeP 2 S 12 (LGPS) and Li 10 GeP 2 O 12 (LGPO) through extended Car-Parrinello molecular dynamics in the canonical and isobaric-isothermal ensemble. The (quasi-)orthorhombic and tetragonal phases are studied both for the oxide and for the sulfide, with the aim of comparing their conductivity with the same approach; out of these four case studies, tetragonal LGPO has not been reported and, while dynamically stable, it sits (0.04 Ha/formula unit) above orthorhombic LGPO. We calculate activation energies for diffusion of 0.18 eV and 0.23 eV for tetragonal and orthorhombic LGPS, and of 0.22 eV and 0.34 eV for tetragonal and orthorhombic LGPO. In line with experiments, we find orthorhombic LGPO orders of magnitude less conductive, at room temperature, than the two sulfide systems. However, this is not the case for tetragonal LGPO, which, although less stable than its orthorhombic allotrope, shows at room temperature a conductivity comparable to orthorhombic and tetragonalLGPS, and, if synthesized, could make a very attractive Li-ion conductor.