For an integer $n \geq 3$, let $M_n$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then $M_n$ is a Fano manifold of complex dimension $n-3$. Let $\Theta_{M_n}$ be the sheaf of germs of holomorphic sections of the tangent bundle $TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all $q \geq 0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on $M_n$ consists of a point. Thus the complex structure on $M_n$ is locally rigid.