High-quality measurements of electromagnetic fields and electron velocity distributions by the Magnetospheric Multiscale (MMS) mission in Earth’s magnetosheath present a unique opportunity to characterize heliospheric plasma turbulence and to determine the mechanisms responsible for its dissipation. We apply the field-particle correlation technique to the MMS measurements to identify the dissipation mechanism and quantify the dissipation rate. It is found that 95% of the intervals have velocity-space signatures of electron Landau damping that are quantitatively consistent with linear kinetic theory for the collisionless damping of kinetic Alfvén waves. About 75% of the intervals have asymmetric signatures, implying that often the collisionless damping is stronger for waves propagating one direction along the magnetic field than the other. Nearly half of the samples have the same electron energization rates as order-of-magnitude estimates of the turbulent energy cascade rate, suggesting that electron Landau damping is frequently responsible for the dissipation of magnetosheath turbulent energy.