We have measured cross sections for forward neutron production from a variety of targets using proton beams from the Fermilab Main Injector. Measurements were performed for proton beam momenta of 58 GeV/c, 84 GeV/c, and 120 GeV/c. The cross section dependence on the atomic weight (A) of the targets was found to vary as $A^(alpha)$ where $\alpha$ is $0.46\pm0.06$ for a beam momentum of 58 GeV/c and 0.54$\pm$0.05 for 120 GeV/c. The cross sections show reasonable agreement with FLUKA and DPMJET Monte Carlos. Comparisons have also been made with the LAQGSM Monte Carlo.