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This work discusses the reachability analysis (RA) of Max-Plus Linear (MPL) systems, a class of continuous-space, discrete-event models defined over the max-plus algebra. Given the initial and target sets, we develop algorithms to verify whether there exist trajectories of the MPL system that, starting from the initial set, eventually reach the target set. We show that RA can be solved symbolically by encoding the MPL system, as well as initial and target sets into difference logic, and then checking the satisfaction of the resulting logical formula via an off-the-shelf satisfiability modulo theories (SMT) solver. The performance and scalability of the developed SMT-based algorithms are shown to clearly outperform state-of-the-art RA algorithms for MPL systems, newly allowing to investigate RA of high-dimensional MPL systems: the verification of models with more than 100 continuous variables shows the applicability of these techniques to MPL systems of industrial relevance.