The present paper is concerned with the numerical integration of linear structural dynamics by means of different higher order accurate Galerkin time integration schemes. Firstly the single field discontinuous and continuous p-Galerkin schemes are developed in a generalized fashion using arbitrary polynomial degrees. In a further step the two field discontinuous and continuous p-Galerkin schemes are derived. In both algorithmic classes continuous Galerkin schemes are obtained by the strong enforcement of the continuity condition as special case of the discontinuous Galerkin schemes. The related time integration schemes are conditioned in an algorithmic set-up in such a manner that the implementation is similar to the classical Newmark scheme and its α derivates. Selected benchmark examples demonstrate the excellent dissipation and dispersion behavior and the robustness of the present Galerkin integration schemes. Furthermore, an error analysis, based on the analytical solution and the real local time integration error, verifies the order of accuracy of Galerkin time integration schemes controlled by the chosen polynomial degree. In particular the order of accuracy for each class of Galerkin integration schemes is specified. The comparison of the numerical effort for the classical Newmark scheme and the family of Galerkin schemes indicates advantages for Galerkin schemes: For a prescribed error level higher order Galerkin schemes are more effective than the Newmark scheme.