The momentum-space signatures of excitons can be experimentally accessed through time-resolved (pump-probe) photoelectron spectroscopy. In this work, we develop a computational framework for exciton photoemission orbital tomography (exPOT) in periodic systems, enabling the simulation and interpretation of experimental observables within many-body perturbation theory. By connecting the $GW$+Bethe-Salpeter equation (BSE) approach to photoemission tomography, our formalism captures exciton photoemission in periodic systems, explicitly incorporating photoemission matrix element effects induced by the light-matter interaction via the probe pulse. The correlated nature of electrons and holes introduces distinct consequences for excitonic photoemission. Using the prototypical two-dimensional material hexagonal boron nitride, we demonstrate these effects, including a dependence of the photoemission angular distribution on the pump pulse polarization. Moreover, our framework extends to excitons with finite center-of-mass momentum, making it well-suited to studying momentum-dark excitons. This provides valuable insights into the microscopic nature of excitonic phenomena in quantum materials.