As far as I know, loss factor is not direction dependent per se, the directional behavior is described by the phase speed phasor (longitudinal, transverse or beam waves).

I consider that the most practical way to calculate the loss factor is by setting up a 3dB testing (or also called Half power method) and test some samples. Basically you will get the damping ratio, which is related to the loss factor as Loss Factor=2*Damping Ratio (According to: https://pdfs.semanticscholar.org/c33d/393514b86fea3045751429361eb4368db99d.pdf). And the directional part will be solved by magnifying the modes that are involved with the direction you want to study.

However, if you want to go further with the directional behavior, I should say that the path you have to follow is to expand the equilibrium equation of momentum by using the Orthotropic constitutive model and obtain a set of equations analogous to the Navier ones for isotropic materials wave propagation. And see how the phase speed phasor is re-defined in those cases.

Another option is to re-take phase velocities defined by Navier equations and assume that you have 3 Young Modulus, 3 Poisson's ratio and 3 Shear modulus instead of just one of them. And adapt existent expressions based on the direction and the type of propagation. That could work as a first approach.