We use one-dimensional particle-in-cell plasma simulations to study the mechanical structure and thermalization properties of collisionless relativistic shock waves in electron-positron plasmas. We consider shocks propagating perpendicularly to the magnetic field direction. We show that these shock waves exist, and that they are completely parameterized by sigma, the ratio of the upstream Poynting flux to the upstream kinetic energy flux. The mechanical properties of the shock structure are controlled by magnetic reflection of all the particles, leading to a strong magnetic overshoot in the leading edge of the shock wave. The shock waves radiate a strong, semicoherent precursor wave with extraordinary mode polarization into the upstream medium, and also create an approximately Rayleigh-Jeans spectrum of extraordinary modes in the downstream medium. The spectrum of these downstream waves is broadband, extending to hundreds of harmonics above the cyclotron frequency based on upstream parameters. The downstream plasma is completely thermalized in the plane perpendicular to the magnetic field direction, with electron and positron distribution functions very close to relativistic Maxwellians, despite the complete absence of Coulomb collisions. The temperatures of these Maxwellian distributions are those expected from the Rankine-Hugoniot shock jump conditions, once these are modified to include the portion of the shock energy invested in extraordinary modes. We show how the jump conditions are modified by the presence of wave fluctuations, and use them to provide a macroscopic description of these collisionless shock flows. We briefly discuss the results of a two-dimensional simulation that demonstrates the generality of our results beyond the assumption of the one-dimensional case. We suggest that the thermalization mechanism is the formation of a synchrotron maser by the coherently reflected particles in the shock front. Because the downstream medium is thermalized, we argue that perpendicular shocks in pure electron-positron plasmas are not good candidates as nonthermal particle accelerators. In related work, Hoshino et al. have shown that relativistic perpendicular shocks in electron-positron-ion plasmas can lead to substantial nonthermal particle acceleration. The shocks in pair plasmas investigated here constitute the leading part of the shock structure in such three-component plasmas. We briefly discuss the possible observability of the high-frequency extraordinary mode radiation created by these shocks. In particular, we suggest that recently observed short time-scale variability of the centimeter wave emission in compact extragalactic radio sources might be attributable to the precursor radiation from relativistic shocks in the magnetized coronae of accretion disks around the central black holes in AGNs.