In this study, the novel fully distributed asynchronous event-triggered protocols that use sampled state information are developed to address the leader-follower and leaderless consensus issues for the general linear multi-agent systems with physically interconnected network. By designing the state estimate of each follower agent, the protocols avoid the requirements for the continuous communication. Meanwhile, this paper proves that the designed protocols don’t exhibit the Zeno behavior in any finite time by giving a strictly positive lower-bound of the minimum inter-event interval. In contrast to the existing related papers, the salient features of this study are that both physically interconnected network and communication network of the multi-agent systems are considered and the developed protocols don’t require global network information, which reduce the conservativeness of previous systems modeling and protocols. Finally, two simulation results are provided to display the performance and effectiveness of the protocols.