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Earthquake damage in historic masonry buildings is generally caused by the collapse of individual portions that become detached form the structure and fail by overturning. These mechanisms are mainly governed by the discrete nature and geometry of the block units. Analyses based on explicit micro-modelling in which each block is considered separately are therefore of increasing importance. This paper offers a robust tool for the seismic assessment of masonry structures under either, quasi-static or dynamic loading. Algorithms for performing dynamic pulses and pushover analyses through the Discrete Element Method are developed and described, taking into account the actual discrete nature and geometry of masonry. A numerical procedure is proposed that automatically detects the collapse mode and follows the evolution of the analysis until collapse. The implemented method is able to give a reliable estimate of the expected failure mechanism, providing the seismic acceleration required to trigger the motion and the ultimate displacement beyond which the collapse occurs. Finally, the implemented algorithms are applied to two case studies and the results are compared with the traditional analysis based on rigid-block kinematics to outline the features and potentialities of the proposed approach.