A procedure for the determination of neutron-induced photon production cross sections is described, including corrections for the neutron-induced photon background and the unfolding of the gamma-ray spectra. As an example, data have been obtained for tantalum and beryllium with five bismuth germanate detectors and the high-energy white neutron source at the Los Alamos Meson Physics Facility. Calculations and measurements indicate that a beryllium sample may be used to simulate a neutron-induced photon background. Results for the Ta(n,xγ) reaction for neutron energies below En = 20 MeV are compared with previous measurements at the Oak Ridge Electron Linear Accelerator.