In this paper we study multi-server tandem queues with finite buffers and blocking after service. The service times are generally distributed. We develop an efficient approximation method to determine performance characteristics such as the throughput and mean sojourn times. The method is based on decomposition into two-station subsystems, the parameters of which are determined by iteration. For the analysis of the subsystems we developed a spectral expansion method. Comparison with simulation shows that the approximation method produces accurate results. So it is useful for the design and analysis of production lines.