Many problems from science and engineering are modeled by Ordinary Differential Equations (ODEs) whose solutions describe the temporal evolution of the modeled processes. In most cases however, the arising equations are too complex to be solved analytically. Consequently, their solutions have to be approximated by numerical methods. In this article, we propose an explicit fourth-derivative two-step linear multistep method (FD2LMM) for ordinary differential equations. The proposed method is constructed by using the maximal order criteria which is obtained through the associated linear difference operator. The starting values used by the proposed method are obtained by suitable single-step method. The order, consistency, linear stability, and the convergence properties of the method are discussed. Numerical experiments are performed and the results are compared with those of existing methods in the literature.