We have used the COBE DIRBE database to derive near- and mid-infrared light curves for a well-defined sample of 38 infrared-bright Mira variable stars and compared with optical data from the AAVSO. In general the 3.5 and 4.9 μm DIRBE bandpasses provide the light curves with the best signal-to-noise ratio (S/N), with S/N decreasing with wavelength at longer wavelengths. At 25 μm good light curves are only available for ~10% of our stars, and at wavelengths >=60 μm extracting high quality light curves is not possible. The amplitude of variability is typically less in the near-infrared than in the optical and less in the mid-infrared than in the near-infrared, with decreasing amplitude with increasing wavelength. On average there are 0.20+/-0.01 mag variation at 1.25 μm and 0.14+/-0.01 mag variation at 4.9 μm for each magnitude variation in V. The observed amplitudes are consistent with results of recent theoretical models of circumstellar dust shells around Mira variables. For a few stars in our sample we find clear evidence of time lags between the optical and near-infrared maxima of phase ~0.05-0.13, with no lags in the minima. For three stars mid-infrared maximum appears to occur slightly before that in the near-infrared, but after optical maximum. We find three examples of secondary maxima in the rising portions of the DIRBE light curves, all of which have optical counterparts in the AAVSO data, supporting the hypothesis that they are due to shocks rather than newly formed dust layers. We find no conclusive evidence for rapid (hours to days) variations in the infrared brightnesses of these stars.