We study weak solutions and its approximation of hyperbolic linear symmetric Friedrichs systems describing acoustic, elastic, or electro-magnetic waves. For the corresponding first-order systems we construct discontinuous Galerkin discretizations in space and time with full upwind, and we show primal and dual consistency. Stability and convergence estimates are provided with respect to a mesh-dependent DG norm which includes the $L_2$ norm at final time. Numerical experiments confirm that the a piori results are of optimal order also for solutions with low regularity, and we show that the error in the DG norm can be closely approximated with a residual-type error indicator.