In this paper, a Cuckoo search algorithm, namely the New Movement Strategy of Cuckoo Search (NMS-CS), is proposed. The novelty is in a random walk with step lengths calculated by Lévy distribution. The step lengths in the original Cuckoo search (CS) are significant terms in simulating the Cuckoo bird's movement and are registered as a scalar vector. In NMS-CS, step lengths are modified from the scalar vector to the scalar number called orientation parameter. This parameter is controlled by using a function established from the random selection of one of three proposed novel functions. These functions have diverse characteristics such as; convex, concave, and linear, to establish a new strategy movement of Cuckoo birds in NMS-CS. As a result, the movement of NMS-CS is more flexible than a random walk in the original CS. By using the proposed functions, NMS-CS achieves the distance of movement long enough at the first iterations and short enough at the last iterations. It leads to the proposed algorithm achieving a better convergence rate and accuracy level in comparison with CS. The first 23 classical benchmark functions are selected to illustrate the convergence rate and level of accuracy of NMS-CS in detail compared with the original CS. Then, the other Algorithms such as Particle Swarm Optimization (PSO), Gravitational Search Algorithm (GSA), and Grey Wolf Optimizer (GWO) are employed to compare with NMS-CS in a ranking of the best accuracy. In the end, three engineering design problems (tension/compression spring design, pressure vessel design and welded beam design) are employed to demonstrate the effect of NMS-CS for solving various real-world problems. The statistical results show the potential performance of NMS-CS in a widespread class of optimization problems and its excellent application for optimization problems having many constraints. Source codes of NMS-CS is publicly available at http://goldensolutionrs.com/codes.html.