  Anisotropic fluid materials are of growing interest with the development of
metamaterials and transformation acoustics. In the general three-dimensional
case, such materials are characterized by a bulk modulus and a full symmetric
matrix of density. Here, a method is presented to retrieve the bulk modulus and
all six components of the density matrix from a selected set of six incident
plane waves impinging on a layer of the material. From the six components of
the density tensor, the three principal directions and the three principal
densities of the material are recovered. The approach relies on the analytical
expression of the reflection and transmission coefficients derived from a state
vector analysis. It results in simple, closed-form, and easily-implementable
inverse relations for the material parameters. As an illustration, the case of
sound propagation through an orthorhombic lattice of overlapping air-filled
ellipsoids is considered, the effective complex and frequency-dependent bulk
modulus and density matrix of which are derived from homogenization cell
problems and account for viscothermal losses. The retrieval method is then
applied to the homogenized layer and results bear testament to its robustness
to extract accurately all seven material parameters. This makes possible the
characterization and design of anisotropic fluid materials in three dimensions.
