Many radionuclei with potential medical applications are difficult to produce routinely, especially those on the proton-rich side of the valley of stability. Current production methods typically use light-ion reactions on materials of similar mass to the target radionuclide, which may require the use of targets poorly suited to withstand irradiation. These reactions may also create significant amounts of proton-rich decay products which require chemical separation from the desired product in a highly radioactive environment. A promising alternative method using heavy-ion fusion-evaporation reactions for the production of the medically relevant radionuclei 52Fe, 76Br and 77Br is presented. Heavy-ion beams of 28Si and 16O were used to bombard natural aluminum, chromium and copper targets at the University of Notre Dame's Nuclear Science Laboratory to produce these and precursor radionuclei by fusion-evaporation reactions. Production yields for these reactions were measured and compared to PACE4 calculations, and simple physical-chemical separation methods which will lead to very high radiopurity yields are proposed. Heavy-ion fusion-evaporation reactions are shown to lead to more reasonable and implementable production, which may scale to satisfy clinical demand. A summary of accelerator facility requirements needed for routine production of these radionuclei is also presented.