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We present a highly accurate and computationally efficient method (herein, the "wavefront orientation method") for determining the travel time of oceanic tsunamis. Based on Huygens' Principle, the method uses an eight-point grid-point pattern and the most recent information on the orientation of the advancing wavefront to determine the time for a tsunami to travel to a specific oceanic location. The method is shown to provide improved accuracy and reduced anisotropy compared with the conventional multiple grid-point method presently in widespread use.