The momentum of phonons should be related to their dispersion relation yes? But one should not say that they have momentum, they have wave vector. But I guess that this falls into your definition of "trivial"
Acoustic phonons do not have mass; that's guaranteed by them being the Goldstone bosons of broken symmetries. Compared with photons, they have much lower speed, so for the same energy, they have vastly greater momentum. In semiconductors, electronic transitions via pure photon processes are called "vertical" because the photon momentum is negligible, whilst phonon processes are called "horizontal" because they're energies are almost negligible.