Inference in matrix-variate Gaussian models has major applications for multi-output prediction and joint learning of row and column covariances from matrix-variate data. Here, we discuss an approach for efficient inference in such models that explicitly account for \iid observation noise. Computational tractability can be retained by exploiting the Kronecker product between row and column covariance matrices. Using this framework, we show how to generalize the Graphical Lasso in order to learn a sparse inverse covariance between features while accounting for a low-rank confounding covariance between samples. We show practical utility on applications to biology, where we model covariances with more than 100,000 dimensions. We find greater accuracy in recovering biological network structures and are able to better reconstruct the confounders.