A solution is presented for the three‐dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on the other five faces. This problem apparently is previously unsolved in the published literature. The Ritz method is used, with displacements assumed in the form of algebraic polynomials. Convergence is studied. Numerical results are given for the first five frequencies of each of the four symmetry classes of vibration, for five thick parallelepiped configurations, including the cube. Contour plots are exhibited for the modal displacements of the cube. The effects of varying Poisson’s ratio are also observed.