This work presents a decoupling multiple-relaxation-time lattice Boltzmann flux solver (MRT-LBFS) for simulating non-Newtonian power-law fluid flows. The decoupling MRT-LBFS is a finite volume solver for the direct update of fluid variables at cell centers. Its fluxes at each cell interface are modeled physically in a mesoscopic way through local reconstruction of the MRT-LBM solutions of density distribution functions (DDFs). In particular, inviscid and viscous fluxes are simultaneously obtained through lattice summations of equilibrium and non-equilibrium DDFs. Following the MRT model, non-equilibrium DDFs are evaluated in the moment space by using the relationships given from the Chapman-Enskog analysis so that collisional invariant properties of conserved variables can be effectively incorporated into the flux reconstruction process. Unlike most existing LB models, in which the relaxation time depends on fluid viscosity, the present method completely decouples the mutual dependence of the relaxation time and viscosity so that the relaxation time can be selected freely. Several numerical examples of non Newtonian power-law fluid flows, including plane Poiseuille flow in a channel, lid-driven cavity flows and polar-cavity flows in a sector, have been simulated for validation. The obtained results compare well with the benchmark data. It has been shown that the decoupling MRT-LBFS has second order of accuracy in space and can be effectively applied to simulate non-Newtonian flows on non-uniform grids. (C) 2016 Published by Elsevier B.V.