We examine interaction mechanisms between the drag-reducing streamwise-traveling waves of spanwise wall velocity applied at the wall of a turbulent channel flow, and near-wall turbulent coherent structures. In particular, we focus our analysis on the drag-reduction-induced modifications of the quasi-streamwise vortices that populate the near-wall region of turbulent flows through a conditional averaging procedure. We observe that, while the streamwise-traveling waves share most of the drag-reducing mechanisms with the spanwise wall oscillations, some important differences related to the finite phase speed of the wave and the different properties of the generalized Stokes layer exist.