Accurate measurements of photonuclear reaction cross sections are crucial for a number of applications, including radiation shielding design, absorbed dose calculations, reactor physics and engineering, nuclear safeguard and inspection, astrophysics, and nuclear medicine. Primarily motivated by the study of the production of selected radionuclides with high-energy photon beams (mainly 225Ac, 47Sc, and 67Cu), we have established a methodology for the measurement of photonuclear reaction cross sections with the microtron accelerator available at the Swiss Federal Institute of Metrology (METAS). The proposed methodology is based on the measurement of the produced activity with a High Purity Germanium (HPGe) spectrometer and on the knowledge of the photon fluence spectrum through Monte Carlo simulations. The data analysis is performed by applying a Bayesian fitting procedure to the experimental data and by assuming a functional trend of the cross section, in our case a Breit-Wigner function. We validated the entire methodology by measuring a well-established photonuclear cross section, namely the 197Au(γ, n)196Au reaction. The results are consistent with those reported in the literature.