In a seminal work, Bertil Matérn introduced several types of processes for modeling repulsive point processes. In this paper an algorithm is presented for the perfect simulation of the Mat´ern III process within a bounded window in Rd fully accounting for edge effects. A simple upper bound on the mean time needed to generate each point is computed when interaction between points is characterized by balls of fixed radius R. This method is then generalized to handle interactions resulting from use of random grains about each point. This includes the case of random radii as a special case. In each case, the perfect simulation method is shown to be provably fast, making it a useful tool for analysis of such processes.