We consider the consensus problem of high-order multi-agent systems, described by multiple integrator dynamics, under general directed graphs. In contrast to the existing results, we propose a fully-distributed consensus algorithm, albeit employing static state feedback. Specifically, it is shown that, with a suitably designed similarity transformation, consensus is reached under only the necessary and sufficient condition of an interconnection graph having a spanning tree. The proposed approach relaxes some restrictive assumptions commonly considered in the available literature, such as imposing global gains in the network and/or exploiting additional information from neighboring agents other than their position-like states. In addition, the proposed approach enables the explicit determination of the final consensus state even in the presence of constant communication delays. Simulation results are provided to illustrate the effectiveness of the proposed approach.