  The results on the inversion of convolution operators as well as Toeplitz
(and block Toeplitz) matrices in the $1$-D (one-dimensional) case are classical
and have numerous applications. Last year, we considered the $2$-D case of
Toeplitz-block Toeplitz (TBT) matrices, described a minimal information, which
is necessary to recover the inverse matrices, and gave a complete
characterisation of the inverse matrices. Now, we develop our approach for the
more complicated cases of block TBT-matrices and $3$-D Toeplitz matrices.
