In this article we describe a novel, phenomenologically based computer simulation approach for studying relaxation dynamics in fluid systems. The method utilizes an ensemble consisting of two isothermal chambers initially separated by an impermeable partition. The fluid configurations in each chamber are initially pre-equilibrated at densities (ρ) over bar+epsilon and (ρ) over bar-epsilon respectively, where (ρ) over bar reflects an average density of interest and epsilon a small perturbation about this value. After the pre-equilibration step the partition is removed and the entire ensemble allowed to relax towards an equilibrium state guided by a kinetic Monte Carlo computer simulation algorithm. Fickian transport coefficients are found from quantities calculated during this relaxation process. We present an analysis of the approach and illustrate its application to transport property calculations in purely diffusive lattice-gas systems. Our results focus upon the critical region for which there are few published results and where simulation results face the most challenges because of finite-size effects and the phenomenon known as critical slowing-down.