We predict a new type of steplike fluctuations in the magnetoconductance of three-dimensional ballistic microwires. These mesoscopic fluctuations, which appear on a scale corresponding to a small fraction of the quantum unit of magnetic flux ${\mathrm{\ensuremath{\Phi}}}_{0}$=hc/e, are a novel manifestation of the Aharonov-Bohm effect. The sharp conductance steps are caused by the shift of electronic levels through the Fermi level in a magnetic field. The flux-induced steps should be observable in, e.g., submicrometer Bi whiskers at temperatures of order 0.1 K. Sample-characteristic fluctuations are predicted to appear in fields as low as a few gauss.