For observers moving with respect to the cosmic rest frame, the microwave background temperature fluctuations will no longer be statistically isotropic. Aside from the familiar temperature dipole, an observer’s velocity will also induce changes in the temperature angular correlation function and create nonzero off-diagonal correlations between multipole moments. We show that both of these effects should be detectable in future full-sky maps from the Planck satellite, and can constrain modifications of the standard cosmological model proposed to explain anomalous current observations.