The deformation spaces of the six orientable 3-dimensional flat Riemannian manifolds are studies. It is proved that the Teichmuller spaces are homeomorphic to the Euclidean spaces. To state more precisely, let $Phi$ denote the holonomy group of the manifold. Then the Teichmuller space is homeomorphic to (1) ${mathbb{R}}^6;if;Phi$ is trivial, (2) ${mathbb{R}}^4;if;Phi$ is cyclic with order two, (3) ${mathbb{R}}^2;if;Phi$ is cyclic of order 3, 4 or 6, and (4) ${mathbb{R}}^3;if;Phi;cong;{mathbb{Z}_2}; imes;{mathbb{Z}_2}$.