We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for multidimensional hyperbolic systems of conservation laws. The strategy is based on the Galilean invariance of dynamic equations and optimization of the reference frame, in which the equations are numerically solved. The optimal reference frame moves (locally in time) with the average characteristic speed of the system, and, in this sense, the resulting method is quasi-Lagrangian. This leads to the acceleration of the numerical computations thanks to the optimal CFL condition and automatic adjustment of the computational domain to the evolving part of the solution. We show that our quasi-Lagrangian acceleration procedure may also reduce the numerical dissipation of the underlying Eulerian method. This leads to a significantly enhanced resolution, especially in the supersonic case. We demonstrate a great potential of the proposed method on a number of numerical examples.