Layered materials can be assembled vertically to fabricate a new class of van der Waals heterostructures a few atomic layers thick, compatible with a wide range of substrates and optoelectronic device geometries, enabling new strategies for control of light-matter coupling. Here, we incorporate molybdenum diselenide/hexagonal boron nitride (MoSe2/hBN) quantum wells in a tunable optical microcavity. Part-light-part-matter polariton eigenstates are observed as a result of the strong coupling between MoSe2 excitons and cavity photons, evidenced from a clear anticrossing between the neutral exciton and the cavity modes with a splitting of 20 meV for a single MoSe2 monolayer, enhanced to 29 meV in MoSe2/hBN/MoSe2 double-quantum wells. The splitting at resonance provides an estimate of the exciton radiative lifetime of 0.4 ps. Our results pave the way for room-temperature polaritonic devices based on multiple-quantum-well van der Waals heterostructures, where polariton condensation and electrical polariton injection through the incorporation of graphene contacts may be realized.