The inertial migration of the elliptical and rectangular particles in a channel flow of a power-law fluid is studied using the lattice Boltzmann method. The numerical method and code are validated by comparing the present results with the previous ones. Effects of power-law index (n) of the fluid, particle shape, particle aspect ratio (a), blockage ratio (k) and Reynolds number (Re) on the particle trajectory and equilibrium position are discussed. The results show that the elliptical and rectangular particles will finally oscillate in a lateral equilibrium position. The distance for the particle from its initial position to the stable equilibrium position is the shortest for the rectangular particle, then followed for the elliptical particle and finally for the circular particle and square particle, the distance is the shortest for the shear-thinning fluid, then followed for the Newtonian fluid and finally for the shear-thickening fluid. This distance for the rectangular particle decreases with increasing Re, but the dependence of the distance on Re for the elliptical particle is not as obvious as that for the rectangular particle. The particle with larger aspect ratio and blockage ratio gets to the equilibrium position faster. The lateral distance from the equilibrium position to the channel centerline is reduced with increasing k, and with decreasing a for both elliptical and rectangular particles. For the particles with larger a, the lateral distance is reduced with the increase of Re, but the relationship between the lateral distance and Re for the particles with smaller a is dependent on n. (C) 2020 Elsevier B.V. All rights reserved.