This technical note proposes a novel approach to design fully distributed reduced-order consensus controllers for multi-agent systems (MASs) with identical general linear dynamics of agents. A new model namely edge dynamics representing the differences on connected agents' states is first presented. Then the distributed consensus controller design is shown to be equivalent to the synthesis of a distributed stabilizing controller for this edge dynamics. Consequently, based on LQR approach, the globally optimal and locally optimal distributed stabilizing controller designs are proposed, of which the locally optimal distributed stabilizing design for the edge dynamics results in a fully distributed consensus controller for the MAS with no conservative bound on the coupling strength. This approach is then further developed to obtain reduced-order distributed consensus controllers for linear MASs. Finally, a numerical example is introduced to demonstrate the theoretical results.