In recent years, the cooperative control of multi-agent systems (MASs) on multi-layer chain networks has become a hot research topic, since the complex connections between layers directly affect the dynamics, propagation, diffusion and consensus of the entire multi-layer chain network. In this paper, the consensus problems for first-order MASs on double-layer and three-layer chain networks are studied, respectively. The eigenvalues of the corresponding system matrices are calculated by means of matrix decomposition first and, through discussing the weight value between agents, some sufficient and necessary conditions for the consensus of MASs on undirected / directed double-layer and three-layer chain networks are established, based on Lyapunov stability theorem, which provide a simple method for judging the consensus of MASs on multi-layer chain networks. Finally, some numerical examples and simulations are presented to confirm the validity of the theoretical results, which lays a foundation for the application of MASs in complex network scenarios.