A new explicit predictor-multicorrector high-order accurate method is suggested for linear elastodynamics. The method is derived from the implicit high-order accurate method based on the time-continuous Galerkin method proposed earlier in our papers. The basic unknowns for the method are displacements and velocities; accelerations are not calculated. The explicit method uses a predictor-multicorrector technique with one or two passes in order to reach the fourth order of accuracy and has controllable numerical dissipation for the suppression of spurious high-frequency oscillations. In contrast to recently suggested explicit high-order accurate methods based on the time-discontinuous Galerkin method, the new method is more accurate (has a higher order of accuracy) and has better algorithmic properties (e.g., a higher-stability limit) at the same computational efforts. Presented numerical examples show the performance of the new method. The method appears to be competitive for medium- and long-term analysis when accuracy of numerical solutions arises an issue due to error accumulation.