We discuss the physics of stochastic particle acceleration in relativistic magnetohydrodynamic (MHD) turbulence, combining numerical simulations of test-particle acceleration in synthetic wave turbulence spectra with detailed analytical estimates. In particular, we study particle acceleration in wavelike isotropic fast mode turbulence, in Alfven and slow Goldreich-Sridhar type wave turbulence (properly accounting for anisotropy effects), including resonance broadening due to wave decay and pitch-angle randomization. At high particle rigidities, the contributions of those three modes to acceleration are comparable to within an order of magnitude, as a combination of several effects (partial disappearance of transit-time damping for fast modes, increased scattering rate for Alfven, and slow modes due to resonance broadening). Additionally, we provide analytical arguments regarding acceleration beyond the regime of MHD wave turbulence, addressing the issue of nonresonant acceleration in a turbulence comprised of structures rather than waves, as well as the issue of acceleration in small-scale parallel electric fields. Finally, we compare our results to the existing literature and provide ready-to-use formulas for applications to high-energy astrophysical phenomenology.