Resonant microelectromechanical Systems (MEMS) have enabled miniaturization of high-performance inertial sensors, radio-frequency filters, timing references and mass-based chemical sensors. Despite the increasing prevalence of MEMS resonators for these applications, the energy dissipation in these structures is not well-understood. Accurate prediction of the energy loss and the resulting quality factor (Q) has significant design implications because it is directly related to device performance metrics including sensitivity for resonant sensors, bandwidth for radio-frequency filters and phase-noise for timing references. In order to assess the future potential for MEMS resonators it is critically important to evaluate the energy dissipation limits, which will dictate the ultimate performance resonant MEMS devices can achieve. This work focuses on the derivation and evaluation of the intrinsic mechanical energy dissipation limit for single-crystal nano and micromechanical resonators due to anharmonic phonon-phonon scattering in the Akhiezer regime. The energy loss is derived using perturbation theory and the linearized Boltzmann transport equation for phonons, and includes the direction and polarization dependent mode-Gruneisen parameters in order to capture the strain-induced anharmonicity among phonon branches. Evaluation of the quality factor limit reveals that Akhiezer damping, previously thought to depend only on material properties, has a strong dependence on crystal orientation and resonant mode shape. The robust model provides a dissipation limit for all resonant modes including shear-mode vibrations, which have significantly reduced energy loss because dissipative phonon-phonon scattering is restricted to volume-preserving phonon branches, indicating that Lame or wine-glass mode resonators will have the highest upper limit on mechanical efficiency. Finally, the analytical dissipation model is integrated with commercial finite element software in order to evaluate the Akhiezer-limited Q for resonators with complex geometries and mode shapes and provide a prediction for the temperature dependence of the energy loss. This theory satisfies the pressing need for a reliable, analytical model that can predict the phonon dissipation limits for modern, resonant MEMS devices, where precise manufacturing techniques and accurate finite element methods can be used to select particular vibrational mode shapes and crystal orientations.