We report on III-nitride waveguides with c-plane GaN/(Al,Ga)N quantum wells in the strong light-matter coupling regime supporting propagating polaritons. They feature a normal-mode splitting as large as 60 meV at low temperatures thanks to the large overlap between the optical mode and the active region, a polariton decay length up to 100 mu m for photonlike polaritons and a lifetime of 1 to 2 ps, with the latter values being essentially limited by residual absorption occurring in the waveguide. The fully lattice-matched nature of the structure allows for very low disorder and high in-plane homogeneity, which is an important asset for the realization of polaritonic integrated circuits that could support nonlinear polariton wave packets up to room temperature thanks to the large exciton binding energy of 40 meV.