Direct numerical simulation of a fully developed turbulent plane Couette-Poiseuille flow with a two-dimensional (2-D) rod-roughened wall is performed to investigate the impacts of the surface roughness. It is shown that the logarithmic region in the mean velocity profile over the rough wall Couette-Poiseuille flow is significantly shortened by the surface roughness compared to that over a turbulent Couette-Poiseuille flow with smooth wall. The Reynolds shear stress over the rough wall Couette-Poiseuille flow is decreased compared to that for a smooth case in the outer layer. These results are attributed to weakened turbulence activity or roll-cell mode over the rough wall Couette-Poiseuille flow near the channel centerline due to suppressed development of u'-structure on the top wall, as documented through spanwise energy spectra of the streamwise velocity fluctuations. Inspection of congregation motion near the bottom wall and time evolution of u'-structure reveal weakened co-supporting cycle for the rough wall case.