Hecke-Kiselman monoids $\textrm{HK}_Θ$ and their algebras $K[\textrm{HK}_Θ]$, over a field $K$, associated to finite oriented graphs $Θ$ are studied. In the case $Θ$ is a cycle of length $n\geqslant 3$, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid $C_n=\textrm{HK}_Θ$ and it is used to describe the structure and the properties of the algebra $K[C_n]$. In particular, it is shown that $K[C_n]$ is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand-Kirillov dimension one. This is used to characterize all Noetherian algebras $K[\textrm{HK}_Θ]$ in terms of the graphs $Θ$. The strategy of our approach is based on the crucial role played by submonoids of the form $C_n$ in combinatorics and structure of arbitrary Hecke-Kiselman monoids $\textrm{HK}_Θ$.