We present a three-dimensional calculation of atmospheric neutrino fluxes using accurate models of the geomagnetic field, hadronic interactions, tracking and decays. Results are presented for the Super-Kamiokande (SK) and Sudbury Neutrino Observatory (SNO) sites using the GEANT-FLUKA hadronic code. We make a comparison with previous one-dimensional calculations and other three-dimensional results. The recently reported geometrical enhancement of low energy, horizontal neutrinos is confirmed. The effect on observed leptons is however, restricted to a small overall increase in flux. We report east-west asymmetries for each of the neutrino species.