Here we predict the existence of a linear bulk spin photovoltaic effect, where spin currents are produced in antiferromagnetic materials as a response to linearly polarized light, and we describe the symmetry requirements for such a phenomenon to exist. This effect does not depend on spin-orbit effects or require inversion symmetry breaking, distinguishing it from previously explored methods. We propose that the physical mechanism is the nonlinear optical effect “shift current,” and calculate from first principles the spin photocurrent for hematite and bismuth ferrite. We predict a significant response in these materials, with hematite being especially promising due to its availability, low band gap, lack of charge photocurrents, and negligible spin-orbit effect.