While the likelihood ratio test (LRT) for the equality of mean vectors when no particular structure is assumed for the covariance matrices is a well-known and well-studied test, the same is not true when some structure, namely a block structure, is assumed for the covariance matrices. In the present work, the author obtains the expressions for the LRT statistics to test the equality of mean vectors when the covariance matrices are assumed to be block-circular or block compound-symmetric and it is shown that actually in most cases the distributions of these statistics have closed finite form representations. For the other cases, families of near-exact distributions are developed and their performance is then numerically assessed. It is shown that these families of near-exact distributions lie very close to the exact distribution, even for very small samples and that they have an asymptotic behavior not only for increasing sample sizes but also for increasing numbers of populations involved and increasing numbers of sets of variables and variables in each set.