Neutrinos with energies above 10(8) GeV are expected from cosmic ray interactions with the microwave background and are predicted in many speculative models. Such energetic neutrinos are difficult to detect, as they are shadowed by Earth, but rarely interact in the atmosphere. Here we propose a novel detection strategy: Earth-skimming neutrinos convert to charged leptons that escape Earth, and these leptons are detected in ground level fluorescence detectors. With the existing HiRes detector, neutrinos from some proposed sources are marginally detectable, and improvements of 2 orders of magnitude are possible at the proposed Telescope Array.