Estimation of covariance matrices when data are missing is a difficult problem. In the multivariate mixed model setting, problems can arise either because the data are unbalanced or because the response vectors are incomplete. Jennrich and Schluchter (1986, Biometrics 42, 805-820) and Laird, Lange, and Stram (1987, Journal of the American Statistical Association 82, 97-105) have proposed EM algorithms for estimating the parameters associated with longitudinal data models. In this paper a new EM algorithm is proposed for the general multivariate mixed model. The algorithm is derivative-free and based on an algorithm for balanced data proposed by Calvin and Dykstra (1991, Annals of Statistics 19, 850-869). Once the algorithm is described it is used to produce estimates of the variance components matrices associated with a completely random two-factor nested model for a bivariate response vector. The data for this example come from Calvin and Sedransk (1991, Journal of the American Statistical Association 86, 36-48), who analyze a study of the quality of care received by cancer patients receiving radiation therapy.