The recent discovery of electrically conductive bacterial appendages has significant physiological, ecological, and biotechnological implications, but the mechanism of electron transport in these nanostructures remains unclear. We here report quantitative measurements of transport across bacterial nanowires produced by the dissimilatory metal-reducing bacterium, Shewanella oneidensis MR-1, whose electron transport system is being investigated for renewable energy recovery in microbial fuel cells and bioremediation of heavy metals and radionuclides. The Shewanella nanowires display a surprising nonlinear electrical transport behavior, where the voltage dependence of the conductance reveals peaks indicating discrete energy levels with higher electronic density of states. Our results indicate that the molecular constituents along the Shewanella nanowires possess an intricate electronic structure that plays a role in mediating transport.