Analytical solutions of the problem on free vibrations of an orthotropic cuboid with free faces, based on the use of trigonometric functions as base ones in all three directions, are constructed. It is established that the solutions obtained describe the purely shearless vibration modes occurring in the body considered at particular ratios between its faces and physicomechanical characteristics of its material. It is shown that, on passing to the zero harmonics in one or two directions, or at zero values of six or all Poisson ratios, the solutions constructed are reduced to the equations corresponding to the classical theory of plates or rods. It is established that, at certain geometrical and physicomechanical parameters, the body considered can have natural frequencies much lower than the known ones.