In this study, a novel fast-implicit iteration scheme called the alternating cell direction implicit (ACDI) method is combined with the approximate factorization scheme. This application aims to offer a mathematically well-defined version of the ACDI method and to increase the accuracy of the iteration scheme used for the numerical solutions of partial differential equations. The ACDI method is a fast-implicit method that can be used for unstructured grids. The use of fast implicit iteration methods with unstructured grids is not common in the literature. The new ACDI method has been applied to the unsteady diffusion equation to determine its convergence and time-dependent solution ability and character. The numerical tests are conducted for different grid types, such as structured, unstructured quadrilateral, and hybrid polygonal grids. Second, the ACDI was applied to the unsteady advection-diffusion equation to understand the time-dependent and progression capabilities of the presented method. Third, a full potential equation solution is created to understand the complex flow solving ability of the presented method. The results of the numerical study are compared with other fast implicit methods, such as the point Gauss-Seidel (PGS) and line Gauss-Seidel (LGS) methods and the fourth-order Runge-Kutta (RK4) method, which is an explicit scheme, and the Laasonen method, which is a fully implicit scheme. The study increased the abilities of the ACDI method. Due to the new ACDI method, the approximate factorization method, which is used only in structural grids that are known to be advantageous, can be applied to any mesh structure.