  Our aim here is to investigate the holomorphic geometric structures on
compact complex manifolds which may not be K\"ahler. We prove that holomorphic
geometric structures of affine type on compact Calabi-Yau manifolds with
polystable tangent bundle (with respect to some Gauduchon metric on it) are
locally homogeneous. In particular, if the geometric structure is rigid in
Gromov's sense, then the fundamental group of the manifold must be infinite. We
also prove that compact complex manifolds of algebraic dimension one bearing a
holomorphic Riemannian metric must have infinite fundamental group.
