The existing results on the leader-following consensus problem for linear continuous-time multi-agent systems over jointly connected switching digraphs rely on the assumption that the system matrices do not have eigenvalues with positive real parts. In this paper, to remove this assumption, we first establish a stability result for a class of linear switched systems. Then, we show that the leader-following consensus problem for linear multi-agent systems with general system modes over jointly connected switching digraphs is solvable if the digraphs are acyclic. Moreover, the leader-following consensus can be achieved at a pre-assigned but arbitrarily fast convergence rate. A numerical example is provided to illustrate our design.