Obtaining the Green tensor for the most general orthotropic medium is not generally possible in a closed form because the solution requires the roots of a sextic, often known as Stroh eigenvalues. The paper gives some conditions under which the sextic can be solved in a closed form for any direction within the space. It enables the construction of classes of orthotropic materials for which the Green tensor can be computed in a closed form (closed-form orthotropic or CFO) for any direction within the space. The cases of transversely isotropic, tetragonal and cubic materials are studied as special cases. The comparison between the exact Green function and approximate Green functions obtained from the nearest CFO material (in the sense of four different distances) is finally performed in the case of five examples of elasticity tensors.