The paper presents a novel effective macro-modelling approach for masonry arches and bridges under cyclic loading, including dynamic actions induced by earthquakes. It utilises an anisotropic material model with embedded discontinuities to represent masonry nonlinearities. Realistic numerical simulations of masonry arch bridges under static and dynamic loading require accurate models representing the anisotropic nature of masonry and material nonlinearity due to opening and closure of tensile cracks and shear sliding along mortar joints. The proposed 3D modelling approach allows for masonry bond via simple calibration, and enables the representation of tensile cracking, crushing and shear damage in the brickwork. A two-scale representation is adopted, where 3D continuum elements at the structural scale are linked to embedded nonlinear interfaces representing the meso-structure of the material. The potential and accuracy of the proposed approach are shown in numerical examples and comparisons against physical experiments on masonry arches and bridges under cyclic static and dynamic loading.