The aim of this paper is to model the viscoelastic properties of micro-cracked materials based on the homogenization micro-macro approach. The isotropic case with random orientation distribution of micro-crack in Burgers non-ageing linear viscoelastic solid was previously modeled. However, in many cases, micro-cracks appear to be parallel and the macroscopic behavior is transversely isotropic. An alternative effective transversely isotropic Burgers model is developed to model the behavior of such materials for the cases of open cracks. For this anisotropic case, the Burgers model for the macroscopic viscoelastic behavior of the micro-cracked materials depends on 20 viscoelastic parameters which are functions of the damage parameter and of the viscoelastic properties of the solid phase. In this study, the evolution of all 20 damage viscoelastic parameters as functions of the crack density is detailed based on a dilute-stress homogenization approach. The use of the same Burgers viscoelastic model as non-cracked material to model the viscoelastic behavior of micro-cracked material is an approximation. This approximation is carried out in short- and long-term behaviors. However, the comparison of this approximation with the exact solution in the case of 1D traction loading showed an excellent validation of this approach in transient situation.