  Let $H$ be a positive semigroup in $\mathbb{Z}^d$ generated by $A$, and let
$K[H]$ be the associated semigroup ring over a field $K$. We investigate
heredity of the Cohen-Macaulay property from $K[H]$ to both its $A$-Newton
graded ring and to its face rings. We show by example that neither one inherits
in general the Cohen-Macaulay property. On the positive side we show that for
every $H$ there exist generating sets $A$ for which the Newton graduation
preserves Cohen-Macaulayness. This gives an elementary proof for an important
vanishing result on $A$-hypergeometric Euler-Koszul homology. As a tool for our
investigations we develop an algorithm to compute algorithmically the Newton
filtration on a toric ring.
