We develop efficient methods for simulating processes of Ornstein-Uhlenbeck type related to the class of $p$-tempered $α$-stable ($\ts$) distributions. Our results hold for both the univariate and multivariate cases and we consider both the case where the $\ts$ distribution is the stationary law and where it is the distribution of the background driving Lévy process (BDLP). In the latter case, we also derive an explicit representation for the transition law as this was previous known only in certain special cases and only for $p=1$ and $α\in[0,1)$. Simulation results suggest that our methods work well in practice.