  We characterize orthonormal bases, Riesz bases and frames which arise from
the action of a countable discrete group $\Gamma$ on a single element $\psi$ of
a given Hilbert space $\mathcal{H}$. As $\Gamma$ might not be abelian, this is
done in terms of a bracket map taking values in the $L^1$-space associated to
the group von Neumann algebra of $\Gamma$. Our result generalizes recent work
for LCA groups. In many cases, the bracket map can be computed in terms of a
noncommutative form of the Zak transform.
