We investigate the effects of geometric order and fluid rheology on the dispersion of micron-sized particles in two-dimensional microfluidic porous media. Particles suspended in a mixture of glycerol and water or in solutions of partially hydrolyzed polyacrylamide (HPAM) polymers were imaged as they flowed through arrays of microscale posts. From the trajectories of the particles, we calculated the velocity distributions and thereafter obtained the longitudinal and transverse dispersion coefficients. Particles flowed in the shear-thinning HPAM solution through periodic arrays of microposts were more likely to switch between streamlines, due to elastic instabilities. As a result, the distributions of particle velocity were broader in HPAM solutions than in glycerol-water mixtures for ordered geometries. In a disordered array of microposts, however, there was little difference between the velocity distributions obtained in glycerol-water and in HPAM solutions. Correspondingly, particles flowed through ordered post arrays in HPAM solutions exhibited enhanced transverse dispersion. This result suggests that periodic geometric order amplifies the effects of the elasticity-induced velocity fluctuations, whereas geometric disorder of barriers effectively averages out the fluctuations.