This paper presents a systematic tuning approach for linear model predictive controllers based on the computationally attractive minimum variance covariance constrained control (MVC3) problem. Unfortunately, the linear feedback policy generated by the MVC3 problem is incompatible with the algorithmic framework of predictive control, in which the primary tuning vehicle is the selection of objective function weights. The main result of this paper is to show that all linear feedbacks generated by the MVC3 problem exhibit the property of inverse optimality with respect to an appropriately defined linear quadratic regulator (LQR) problem. Thus, the proposed tuning scheme is a two-step procedure: application of the MVC3 problem to achieve tuning objectives, followed by application of inverse optimality to determine the predictive control weights from the MVC3 -generated linear feedback.