An electron beam traversing a structured plasmonic field is shown to undergo diffraction with characteristic angular patterns of both elastic and inelastic outgoing electron components. In particular, a plasmonic grating (e.g., a standing wave formed by two counterpropagating plasmons in a thin film) produces diffraction orders of the same parity as the net number of exchanged plasmons. Large diffracted beam fractions are predicted to occur for realistic plasmon intensities in attainable geometries due to a combination of phase and amplitude changes locally imprinted on the passing electron wave. Our study opens vistas in the study of multiphoton exchanges between electron beams and evanescent optical fields with unexplored effects related to the transversal component of the electron wave function.