For discretized linear periodic heating problems, the nodal temperatures at the end of a heating cycle must be a linear function of those at the beginning of the same cycle and of any reservoirs with which the system exchanges heat. If the matrix of coefficients defining this system of linear equations is determined, it can be applied sequentially to advance through the transient response to the system without the need for direct numerical simulation of every cycle. It also permits a direct solution for the nodal temperatures at an appropriate time during the steady periodic state. Methods of determining the required matrices are discussed and the method is applied to a simple example.