  Let $S$ be the power series ring or the polynomial ring over a field $K$ in
the variables $x_1,\ldots,x_n$, and let $R=S/I$, where $I$ is proper ideal
which we assume to be graded if $S$ is the polynomial ring. We give an explicit
description of the cycles of the Koszul complex whose homology classes generate
the Koszul homology of $R=S/I$ with respect to $x_1,\ldots,x_n$. The
description is given in terms of the data of the free $S$-resolution of $R$.
The result is used to determine classes of Golod ideals, among them proper
ordinary powers and proper symbolic powers of monomial ideals. Our theory is
also applied to stretched local rings.
