We present results from three-dimensional hybrid-kinetic simulations of Alfv\'enic turbulence in a high-beta, collisionless plasma. The key feature of such turbulence is the interplay between local wave-wave interactions between the fluctuations in the cascade and the non-local wave-particle interactions associated with kinetic micro-instabilities driven by anisotropy in the thermal pressure (namely, firehose, mirror, and ion-cyclotron). We present theoretical estimates for, and calculate directly from the simulations, the effective collisionality and plasma viscosity in pressure-anisotropic high-beta turbulence, demonstrating that, for strong Alfv\'enic turbulence, the effective parallel viscous scale is comparable to the driving scale of the cascade. Most of the cascade energy (80-90%) is dissipated as ion heating through a combination of Landau damping and anisotropic viscous heating. The kinetic-energy spectrum of the turbulence has a slope steeper than
to the anisotropic viscous stress. The magnetic-energy spectrum is shallower
than near the ion-Larmor scale due to fluctuations produced by the
firehose instability. Our results have implications for models of particle
heating in low-luminosity accretion onto supermassive black holes, the
effective viscosity of the intracluster medium, and the interpretation of
near-Earth solar-wind observations.