We consider the blind separation of sources with general (e.g., not necessarily stationary) temporal covariance structures. When the sources' temporal covariance matrices are known, the maximum-likelihood (ML) separation scheme (for Gaussian sources) conveniently exploits this knowledge. However, in the more practical case, when these matrices are unknown, ML separation calls for their estimation from the available observations. When multiple snapshots of the mixtures are available (synchronized to some external stimulus), such estimation is possible, but might require a huge number of snapshots for attaining reasonable accuracy. Rather than estimate high-dimensional covariance matrices, we propose here a more practical ("partial"-ML) approach, based on estimation of much smaller covariance matrices. These are covariances of low-dimensional vectors, consisting of respective off-diagonal terms of spatial sample-correlation matrices. Weighted joint diagonalization of these correlation matrices (using the estimated low-dimensional covariances for the weighting) significantly improves the separation performance over alternative options, as we demonstrate in simulation.