We show that the classical homogenization is able to describe the dispersion relation of spoof plasmons in structured thick interfaces with periodic unit cell being at the subwavelength scale. This is because the interface in the real problem is replaced by a slab of an homogeneous birefringent medium, with an effective mass density tensor and an effective bulk modulus. Thus, explicit dispersion relation can be derived, corresponding to guided waves in the homogenized problem. Contrary to previous effective medium theories or retrieval methods, the homogenization gives effective parameters depending only on the properties of the material and on the geometry of the microstructure. Although resonances in the unit cell cannot be accounted for within this low-frequency homogenization, it is able to account for resonances occurring because of the thickness of the interface and thus, to capture the behaviour of the spoof plasmons. Beyond the case of simple grooves in a hard material, we inspect the influence of tilting the grooves and the influence of the material properties.