  We consider semi-direct products $\C^{n}\ltimes_{\phi}N$ of Lie groups with
lattices $\Gamma$ such that $N$ are nilpotent Lie groups with left-invariant
complex structures. We compute the Dolbeault cohomology of direct sums of
holomorphic line bundles over $G/\Gamma$ by using the Dolbeaut cohomology of
the Lie algebras of the direct product $\C^{n}\times N$. As a corollary of this
computation, we can compute the Dolbeault cohomology $H^{p,q}(G/\Gamma)$ of
$G/\Gamma$ by using a finite dimensional cochain complexes. Computing some
examples, we observe that the Dolbeault cohomology varies for choices of
lattices $\Gamma$.
