This paper proposes an inverse estimation method for the characterisation of the elastic and anelastic properties of the frame of anisotropic open-cell foams. A model of viscoelasticity based on a fractional differential constitutive equation is used, leading to an augmented Hooke’s law in the frequency domain, where the elastic and anelastic phenomena appear as distinctive terms in the stiffness matrix. The model parameters consist of nine orthotropic elastic moduli, three angles of orientation of the material principal directions and three parameters governing the anelastic frequency dependence. The inverse estimation consists in numerically fitting the model on a set of transfer functions extracted from the sample. For this purpose, a seismic-mass setup is repeated in the three directions of space and placed in a vacuum chamber in order to remove the air from the sample. The method allows to reconstruct the full frequency-dependent complex stiffness matrix of the frame of an anisotropic open-cell foam and in particular it provides the frequency of maximum energy dissipation by viscoelastic effects. The characterisation of a melamine foam sample is performed and the relation between the fractional-derivative model and other types of parametrisations of the augmented Hooke’s law is discussed.