Theoretical and numerical analyses and some observations on the physical phenomena of free vibration of solid cylinders having square and hexagonal cross-sections with combinations of fixed and free ends are reported. The formulation is established based on the linear, small-strain, three-dimensional elasticity principle, and the p-Ritz method is employed for computing the solution of the problem. By expanding the displacements in spatial co-ordinates, integral expressions for strain and kinetic energies in a three-dimensional setting are derived in Cartesian form. Sets of one-and two-dimensional orthogonal polynomials are constructed to represent the three-dimensional variations in the longitudinal and lateral surface directions. A basic function is introduced in these polynomials to cater for the stress free lateral surfaces and the kinematic constraints at both ends. Several examples are solved to demonstrate the applicability of the method. First known results in terms of vibration frequency parameters and mode shapes of cylinders having square and hexagonal cross-sections for various symmetry classes and end constraints are presented and the physics behind the results is highlighted. These new results may serve as benchmark data for future research development in simplified beam theories.