  The algebra of $n\times n$ matrices over a field $F$ has a natural
$\mathbb{Z}_n$-grading. Its graded identities have been described by Vasilovsky
who extended a previous work of Di Vincenzo for the algebra of $2\times 2$
matrices. In this paper we study the graded identities of block-triangular
matrices with the grading inherited by the grading of $M_n(F)$. We show that
its graded identities follow from the graded identities of $M_n(F)$ and from
its monomial identities of degree up to $2n-2$. In the case of blocks of sizes
$n-1$ and 1, we give a complete description of its monomial identities, and
exhibit a minimal basis for its $T_{\mathbb{Z}_n}$-ideal.
