The vibrations of gold nanowires and nanorods are investigated numerically in the framework of continuum elasticity using the Rayleigh-Ritz variational method. Special attention is paid to identify the vibrations relevant in Raman scattering experiments. A comprehensive description of the vibrations of nanorods is proposed by determining their symmetry, comparing with standing waves in the corresponding nanowires, and estimating their Raman intensity. The role of experimentally relevant parameters such as the anisotropic cubic lattice structure, the presence of faceted lateral surfaces, and the shape of the ends of the nanorods is evaluated. Elastic anisotropy is shown to play a significant role contrarily to the presence of facets. Localized vibrations are found for nanorods with flat ends. Their evolution as the shape of the ends is changed to half-spheres is discussed.