We demonstrate with experiments and simulations how microscopic self-propelled particles navigate through environments presenting complex spatial features, which mimic the conditions inside cells, living organisms and future lab-on-a-chip devices. In particular, we show that, in the presence of periodic obstacles, microswimmers can steer even perpendicularly to an applied force. Since such behaviour is very sensitive to the details of their specific swimming style, it can be employed to develop advanced sorting, classification and dialysis techniques.