We compute the effective wavefront speeds of reaction-diffusion equations in periodically layered media with coefficients that have small-amplitude oscillations around a uniform mean state. We compare them with the corresponding wavefront speeds in the uniform state. We analyze a one-dimensional model where wave propagation is along the layering direction of the medium and a two-dimensional shear flow model where wave propagation is othogonal to the layering direction. We find that the effective wave speed is smaller in the one-dimensional model and is larger in the two-dimensional model for both bistable cubic and quadratic nonlinearities of the Kolmogorov-Petrovskii-Piskunov form. We derive approximate expressions for the wave speeds in the bistable case.