We present predictions for the reionization of the intergalactic medium (IGM) by stars in high-redshift galaxies, based on a semi-analytic model of galaxy formation. We calculate ionizing luminosities of galaxies, including the effects of absorption by interstellar gas and dust on the escape fraction ƒesc, and follow the propagation of the ionization fronts around each galaxy in order to calculate the filling factor of ionized hydrogen in the IGM. For a ΛCDM cosmology, with parameters of the galaxy formation model chosen to match observations of present-day galaxies, and a physical calculation of the escape fraction, we find that the hydrogen in the IGM will be reionized at redshift z=6.1 if the IGM has uniform density, but only by z=4.5 if the IGM is clumped. If instead we assume a constant escape fraction of 20 per cent for all galaxies, then we find reionization at z=4.5 and 7.8 for the same two assumptions about IGM clumping. We combine our semi-analytic model with an N-body simulation of the distribution of dark matter in the Universe in order to calculate the evolution of the spatial and velocity distribution of the ionized gas in the IGM, and use this to calculate the secondary temperature anisotropies induced in the cosmic microwave background (CMB) by scattering off free electrons. The models predict a spectrum of secondary anisotropies covering a broad range of angular scales, with fractional temperature fluctuations ∼10−7-10−6 on arcminute scales. The amplitude depends strongly on the total baryon density, and less sensitively on ƒesc. The amplitude also depends somewhat on the geometry of reionization, with models in which the regions of highest gas density are reionized first giving larger CMB fluctuations than the case where galaxies ionize surrounding spherical regions, and models where low-density regions reionize first giving the smallest fluctuations. Measurement of these anisotropies can therefore put important constraints on the reionization process, in particular, the redshift evolution of the filling factor, and should be a primary objective of a next generation submillimetre telescope such as the Atacama Large Millimeter Array.