This paper investigates sampled-data consensus problems of general linear multi-agent systems under switching topologies. To perform the consensus analysis by the averaging method, the multi-agent system in the sampled-data setting is first converted into the continuous-time system with the input delay. Then, the approximate relationship between the states of this input delay system and the states of its averaged system is established by using the tool from non-standard analysis. Subsequently, a sufficient condition ensuring the sampled-data consensus is obtained by linear matrix inequalities. It is shown that the sampled-data consensus of general linear multi-agent systems, including exponential unstable agents, can be achieved, if the speed of topology switching is fast enough and the sampled-data consensus under the averaging topology can be achieved. Finally, numerical simulations are provided to demonstrate the effectiveness of theoretical analyses.