Here we propose that the radioresistance (tolerance to ionizing radiation) observed in several terrestrial bacteria has a martian origin. Multiple inconsistencies with the current view of radioresistance as an accidental side effect of tolerance to desiccation are discussed. Experiments carried out 25 years ago were reproduced to demonstrate that "ordinary" bacteria can develop high radioresistance ability after multiple cycles of exposure to high radiation dosages followed by cycles of recovery of the bacterial population. We argue that "natural" cycles of this kind could have taken place only on the martian surface, and we hypothesize that Mars microorganisms could have developed radioresistance in just several million years' time and, subsequently, have undergone transfer to Earth by way of martian meteorites. Our mechanism implies multiple and frequent exchanges of biota between Mars and Earth.