Aims. The main goal of this work is to measure rotation periods of the M-type dwarf stars being observed by the CARMENES exoplanet survey to help distinguish radial-velocity signals produced by magnetic activity from those produced by exoplanets. Rotation periods are also fundamental for a detailed study of the relation between activity and rotation in late-type stars. Methods. We look for significant periodic signals in 622 photometric time series of 337 bright, nearby M dwarfs obtained by long-time baseline, automated surveys (MEarth, ASAS, SuperWASP, NSVS, Catalina, ASAS-SN, K2, and HATNet) and for 20 stars which we obtained with four 0.2–0.8 m telescopes at high geographical latitudes. Results. We present 142 rotation periods (73 new) from 0.12 d to 133 d and ten long-term activity cycles (six new) from 3.0 a to 11.5 a. We compare our determinations with those in the existing literature; we investigate the distribution of P_rot in the CARMENES input catalogue, the amplitude of photometric variability, and their relation to v sini and pEW(Hα); and we identify three very active stars with new rotation periods between 0.34 d and 23.6 d.