  In this paper we investigate the existence of invariant SKT, balanced and
generalized K\"ahler structures on compact quotients $\Gamma \backslash G$,
where $G$ is an almost nilpotent Lie group whose nilradical has one-dimensional
commutator and $\Gamma$ is a lattice of $G$. We first obtain a characterization
of Hermitian almost nilpotent Lie algebras $\mathfrak{g}$ whose nilradical
$\mathfrak{n}$ has one-dimensional commutator and a classification result in
real dimension six. Then, we study the ones admitting SKT and balanced
structures and we examine the behaviour of such structures under flows. In
particular, we construct new examples of compact SKT manifolds. Finally, we
prove some non-existence results for generalized K\"ahler structures in real
dimension six. In higher dimension we construct the first examples of non-split
generalized K\"ahler structures (i.e., such that the associated complex
structures do not commute) on almost abelian Lie algebras. This leads to new
compact (non-K\"ahler) manifolds admitting non-split generalized K\"ahler
structures.
