This paper describes a novel linearly-weighted gradient smoothing method (LWGSM). The proposed method is based on irregular cells and thus can be used for problems with arbitrarily complex geometrical boundaries. Based on the analysis about the compactness and the positivity of coefficients of influence of their stencils for approximating a derivative, one favorable scheme (VIII) is selected among total eight proposed discretization schemes. This scheme VIII is successively verified and carefully examined in solving Poisson equations, subjected to changes in the number of nodes, the shapes of cells and the irregularity of triangular cells, respectively. Strong form of incompressible Navier-Stokes equations enhanced with artificial compressibility terms are tackled, in which the spatial derivatives are approximated by consistent and successive use of gradient smoothing operation over smoothing domains at various locations. All the test cases using LWGSM solver exhibits its robust, stable and accurate behaviors. The attained incompressible LWGSM solutions show good agreements with experimental and literature data. Therefore, the proposed LWGSM can be reliably used for accurate solutions to versatile fluid flow problems.