We describe spatio-temporal random processes using linear mixed models and discuss estimation, inference, and prediction under this formulation. We show how many commonly used covariances are special cases of this more general framework and pay special attention to the separable and product-sum covariances. The linear mixed model formulation facilitates straightforward implementation of computationally efficient inversion of separable and product-sum covariance matrices, even when every spatial location is not observed at every time point. Through a simulation study, we find that for many configurations of Gaussian data, likelihood-based estimation tends to outperform semivariogram-based estimation with respect to fixed effect and prediction performance. We also use this simulation study to identify scenarios where separable covariances are noticeably inferior to product-sum covariances with respect to these performance metrics. We also analyze daily maximum temperature at various locations in Oregon, USA, during the 2019 summer and evaluate inferential conclusions and prediction performance for these covariances and estimation methods; we find the product-sum covariance estimated using restricted maximum likelihood performs best. We then provide guidelines for choosing among combinations of these covariances and estimation methods based on properties of observed data and end with a summary and discussion of future work.