Consider a non-symmetric generalized diffusion X(.) in R-d determined by the differential operator A(x) = - Sigma(ij)partial derivative(i)a(ij)(x)partial derivative(j) + Sigma(i)b(i)(x)partial derivative(i). In this paper the diffusion process is approximated by Markov jump processes X-n(.), in homogeneous and isotropic grids G(n) subset of R-d, which converge in distribution in the Skorokhod space D([0, infinity), R-d) to the diffusion X(.). The generators of X-n(.) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for d >= 3 can be applied to processes for which the diffusion tensor {a(ij)(x)}(11)(dd) fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes X-n(.). For piece-wise constant functions a(ij) on R-d and piece-wise continuous functions a(ij) on R-2 the construction and principal algorithm are described enabling an easy implementation into a computer code.