We report on the results of three-dimensional calculations of thermohydrologic behavior on a planet with a frozen surface heated from below for ranges of model parameters relevant to Mars. Our results indicate that adopting a more realistic model (i.e., three-dimensional convective view in contrast to a one-dimensional static view) of the Martian subsurface significantly changes the possibilities with respect to the depth at which liquid water might exist on Mars. These calculations suggest that hydrothermal convection may occur in the Martian regolith, driven solely by background geothermal heating, and that the sustained upwelling of warm convecting fluids may significantly thin the permafrost ice layer on Mars at some locations, to as thin as 300 m for certain parameter choices.