Raman spectroscopy is a powerful analytical method widely used in many fields of science and applications. However, one of the inherent issues of this method is a low signal-to-noise ratio for ultrathin and two-dimensional (2D) materials. To overcome this problem, techniques like surface-enhanced Raman spectroscopy (SERS) that rely on nanometer scale metallic particles are commonly employed. Here, we demonstrate a different approach that is based on a microcavity structure consisting of a hexagonal boron nitride (h-BN) membrane spanning over an air-filled trench in germanium. In this structure, the h-BN membrane is an integral part of the cavity and, at the same time, shows an about 10-fold, polarisation-dependent h-BN Raman signal enhancement. With h-BN being transparent, flat, and chemically robust, it provides an excellent interface between the cavity and adjacent materials. We show that the Raman enhancement is also present for graphene layers transferred on top of the h-BN membrane, which proves that our approach can be extended to van der Waals heterostructures. The observed polarisation and position-dependent enhancements are in very good agreement with numerical simulations of the electric field intensity of the cavity. These results, together with the presented facile h-BN membrane fabrication process, which does not require any lithographic methods, open up new possibilities for enhancing Raman signals of 2D crystals without the need for metal particles.