  We give a combinatorial description of local cohomology modules of a graded
module over a semigroup ring, with support at the graded maximal ideal. This
combinatorial framework yields Hochster-type formulas for the Hilbert series of
such local cohomology modules in terms of the homology of finitely many
polyhedral cell complexes. A Cohen--Macaulay criterion immediately follows. We
also provide an alternative proof of a result of [18] characterizing
Cohen--Macaulay affine semigroup rings.
