  Let $K[HK_{\Theta}]$ denote the Hecke-Kiselman algebra of a finite oriented
graph $\Theta$ over an algebraically closed field $K$. All irreducible
representations, and the corresponding maximal ideals of $K[HK_{\Theta}]$, are
characterized in case this algebra satisfies a polynomial identity. The latter
condition corresponds to a simple condition that can be expressed in terms of
the graph $\Theta$. The result shows a surprising similarity to the classical
results on representations of finite semigroups; namely every representation
either comes form an idempotent in the Hecke-Kiselman monoid $HK_{\Theta}$ (and
hence it is $1$-dimensional), or it comes from certain semigroup of matrix type
(which is an order in a completely $0$-simple semigroup over an infinite cyclic
group). The case when $\Theta$ is an oriented cycle plays a crucial role; the
prime spectrum of $K[HK_{\Theta}]$ is completely characterized in this case.
