A new analytical methodology has been developed to determine the diagonal and cross-diagonal molecular diffusion coefficients in ternary mixtures by the Sliding Symmetric Tubes technique. The analytical solution is tested in binary mixtures obtaining good agreement with the results of the literature. Results are presented for the ternary mixture formed by tetralin, isobutylbenzene, and dodecane with an equal mass fraction for all the components (1-1-1) which is held at 25 °C. Diagonal and cross-diagonal coefficients are determined for the three possible orders of components, in order to compare the results with those available in the literature. A comparison with published results shows a good agreement for the eigenvalues of the diffusion matrix, and a reasonable agreement for the diagonal molecular diffusion coefficients.