We investigate consensus dynamics in networks of coupled discrete-time linear dynamical systems in which agents anticipate the future states of its neighbors using past information. This anticipation mechanism introduces a delay term and incorporates memory into the consensus protocol. We analyze how anticipation affects the stability and the speed of consensus. In particular, we derive necessary and sufficient conditions under which anticipation leads to faster convergence compared to the undelayed case. On the other hand, the delays introduced by anticipation can also break consensus altogether under certain other conditions. Our analysis relies on the spectral properties of the graph Laplacian and shows that the convergence rate depends on the smallest nonzero and the largest eigenvalues of the Laplacian.