The cluster randomized crossover design has been proposed to improve efficiency over the parallel design with a limited number of clusters. In recent years, the cluster randomized crossover design has been increasingly used in evaluating the effectiveness of health care policy or programs. Since interest often lies in quantifying the population-averaged intervention effect in these studies, we develop sample size procedures for continuous and binary outcomes corresponding to a population-averaged model estimated by the generalized estimating equations (GEE), accounting for both the within-period and inter-period correlations. We show that the required sample size depends on the correlation parameters through an eigenvalue of the within-cluster correlation matrix for continuous outcomes and through two distinct eigenvalues of the correlation matrix for binary outcomes. We demonstrate that the empirical power corresponds well with the predicted power by the proposed formula for as few as 8 clusters, when outcomes are analyzed using a bias-corrected estimating equations for the correlation parameters concurrently with a suitable bias-corrected sandwich variance estimator.