We monitored the light variations of 58 weak-line T Tauri stars in Taurus, recently discovered in the X-ray wavelength range during the ROSAT All-Sky Survey. We derive photometric periods for 18 stars, all but one being ascribed to rotational modulation by stellar spots. The exception is a 37.6d period assigned to the orbital motion of a new pre-main sequence spectroscopic binary. Two thirds of the stars in our sample have an age larger than 10Myr and up to 40Myr, thus filling the observational gap that previously existed between T Tauri stars on convective tracks and ZAMS dwarfs for the determination of the rotational evolution of young low-mass stars. The rotational periods are found to range from 0.5 to 7.5-days, most periods being shorter than 5-days. This result provides direct evidence for the spin up of solar type stars as they contract on pre-main sequence radiative tracks, as predicted by recent models of angular momentum evolution. The paucity of long periods (P>=5d, i.e., V_eq_<=10km/s) in the sample of post-T Tauri stars leaves, however, the origin of the numerous slow rotators observed in young clusters an open issue.