Existing consensus control algorithms for networked discrete-time linear systems have slow convergence to consensus trajectories due to limitations on the magnitude of the consensus gain. In this work we propose and analyze a new predictor-based consensus control protocol that recovers the positive features of the continuous-time counterpart in terms of arbitrary rate of convergence to the consensus dynamics. Moreover, it admits a distributed implementation based on multiple consensus steps. Our analysis concerns general weakly connected digraphs and it encompasses both single and multiple consensus.