We discuss the computational complexity of the multidimensional periodic scheduling problem. This problem originates from the assignment of periodic tasks to processing units over time and it is related to the design of high-performance video signal processors. We present a model of multidimensional periodic operations and introduce the multidimensional periodic scheduling problem. Next, we show that this problem and two related sub-problems are NP-hard. Further-more, we identify several special cases induced by practical situations, of which some are proven to be polynomially solvable.