The class of polling systems, i.e., multiqueue systems with cyclic service, plays an important role in the performance evaluation of various computer and communication systems, e.g., switching systems with distributed control and token-passing local-area networks. Although the behavior of users and the corresponding incoming traffic characteristics in such systems are increasingly complex, most analytical performance studies in the literature are based on the assumption of Poissonian input processes. In this paper, an approximate discrete-time analysis of polling systems with finite capacity of waiting places and nonexhaustive service is presented, considering general renewal input traffic. The analysis method is based on the evaluation of discrete convolution operations using fast convolution algorithms, e.g., the Fast Fourier Transform (FFT). To illustrate the accuracy of the approximation and its dependency on system parameters, numerical results are given.