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We present a calculation of the production of neutrinos during propagation of ultra-high energy cosmic rays from their astrophysical sources to us. Photoproduction interactions are modeled with the event generator SOPHIA that represents very well the experimentally measured particle production cross sections at accelerator energies. We give the fluxes expected from different assumptions on cosmic ray source distributions, cosmic ray injection spectra, cosmological evolution of the sources and different cosmologies, and compare them to the Waxman-Bahcall limit on source neutrinos. We estimate rates for detection of neutrino induced showers in a km3 water detector. The ratio of the local high energy neutrino flux to the ultra-high energy cosmic ray flux is a crucial parameter in distinguishing between astrophysical and cosmological (top-down) scenarios of the ultra-high energy cosmic ray origin.