In this paper, we propose a new approach to the design of training sequences that can be used for an accurate estimation of multi-input multi-output channels. The proposed method is particularly instrumental in training sequence designs that deal with three key challenges: 1) arbitrary channel and noise statistics that do not follow specific models, 2) limitations on the properties of the transmit signals, including total power, per-antenna power, having a constant-modulus, discrete-phase, or low peak-to-average-power ratio, and 3) signal design for large-scale or massive antenna arrays. Several numerical examples are provided to examine the proposed method.