In longitudinal studies, repeated measures are collected over time and hence they tend to be serially correlated. In this paper we consider an extension of skew-normal/independent linear mixed models introduced by Lachos et al. (2010), where the error term has a dependence structure, such as damped exponential correlation or autoregressive correlation of order p. The proposed model provides flexibility in capturing the effects of skewness and heavy tails simultaneously when continuous repeated measures are serially correlated. For this robust model, we present an efficient EM-type algorithm for computation of maximum likelihood estimation of parameters and the observed information matrix is derived analytically to account for standard errors. The methodology is illustrated through an application to schizophrenia data and several simulation studies. The proposed algorithm and methods are implemented in the new R package skewlmm.