This paper investigates the leader-following consensus problem for general linear multi-agent systems with time-varying delay. An observer-based distributed triggering mechanism is proposed to solve this problem. For each agent, the update of the designed distributed event-triggered controller only requires the knowledge of its neighbors' states at its own event time instants. A sufficient condition is presented to show that leader-following consensus of the multi-agent system can be reached exponentially. Moreover, the Zeno behavior of the triggering time sequences is excluded naturally. Finally, the effectiveness of the proposed control schemes is illustrated by a numerical example.