A new high-order accurate time-continuous Galerkin (TCG) method for elastodynamics is suggested. The accuracy of the new implicit TCG method is increased by a factor of two in comparison to that of the standard TCG method and is one order higher than the accuracy of the standard time-discontinuous Galerkin (TDG) method at the same number of degrees of freedom. The new method is unconditionally stable and has controllable numerical dissipation at high frequencies. An iterative predictor/multi-corrector solver that includes the factorization of the effective mass matrix of the same dimension as that of the mass matrix for the second-order methods is developed for the new TCG method. A new strategy combining numerical methods with small and large numerical dissipation is developed for elastodynamics. Simple numerical tests show a significant reduction in the computation time (by 5-25 times) for the new TCG method in comparison to that for second-order methods, and the suppression of spurious high-frequency oscillations.