A solution for the calculation of three-dimensional (3D) eigenrays based on Simplex optimization, implemented in a 3D Gaussian beam model, is investigated in this paper. The validation and performance of the solution were analyzed through comparisons against an equivalent (flat) two-dimensional waveguide, and against results of a tank scale experiment presented in Sturm and Korakas [(2013). J. Acoust. Soc. Am. 133(1), 108-118], in which cross-slope propagation in a wedge waveguide with a mild slope was considered. It was found that the search strategy based on Simplex optimization was able to calculate efficiently and accurately 3D eigenrays, thus providing predictions of arrival patterns along cross-slope range, which replicated elaborate patterns of mode shadow zones, intra-mode interference, and mode arrivals. A remarkable aspect of the search strategy was its ability to provide accurate values of initial eigenray elevation and azimuth, within the accuracy defined for the eigenray to arrive at the location of a given hydrophone.