We examine a coherently-driven, dispersion-managed, passive Kerr fiber ring resonator and report the first direct experimental observation of dispersive wave emission by temporal cavity solitons. Our observations are in excellent agreement with analytical predictions and they are fully corroborated by numerical simulations. These results lead to a better understanding of the behavior of temporal cavity solitons under conditions where higher-order dispersion plays a significant role. Significantly, since temporal cavity solitons manifest themselves in monolithic microresonators, our results are likely to explain the origins of spectral features observed in broadband Kerr frequency combs.