We present an ab initio computational study of triplet exciton diffusion in four phosphorescent emitters commonly used in organic light-emitting diodes (OLEDs). By kinetic Monte Carlo simulations, triplet diffusion lengths are obtained for these emitters in neat films and as a guest in two different hosts. The triplet transfer rates governing the diffusion contain a transfer integral factor that includes both Förster and Dexter contributions and a Franck-Condon weighted density of vibrational states that includes the coupling to all intramolecular vibrations in a fully quantum mechanical way. We find that at guest concentrations around 10 mol% the Förster transfer contribution is most important. At larger concentrations of about 30-40 mol% the Dexter contribution becomes dominant. We show that obtaining the triplet transfer rates by the semiclassical Marcus theory yields diffusion lengths that are too short and that using a simple cubic lattice in combination with the often used Miller-Abrahams rates instead of using a real morphology with the ab initio rates leads to an underestimation of the diffusion lengths due to transfers down in energy that are too slow.