We calculate the radio frequency (RF) spectrum of fermionic atoms near a narrow Feshbach resonance, explaining observations made in ultracold samples of 6Li. We use a two channel resonance model to show that the RF spectrum contains two peaks. In the wide-resonance limit, nearly all spectral weight lies in one of these peaks, and typically the second peak is very broad. We find strong temperature dependence, which can be traced to the energy dependence of the two-particle scattering. In addition to microscopic calculations, we use sum rule arguments to find generic features of the spectrum which are model independent. © 2013 American Physical Society.