In Vienna, about one year ago, researchers proposed that a previously-discovered meson is the glueball, a massive particle that consists of massless gluons (this is their published paper in Phys. Rev. Lett). Can't the same mechanism be responsible for the mass of quarks, leptons and other massive particles? If so, they have to be composites of massless particles, of course, so maybe this discovery is at the same time a hint that that's indeed the case (as in the Rishon theory of Harari).
In the case of glueballs the large majority of the mass is not caused by the Higgs mechanism but from the potential energy that binds the gluons together (remember $E = mc^2$). A similar mechanism gives hadrons and mesons (bound states of quarks and gluons) the majority of their mass. See for example this threat.
Quarks and other leptons themselves are (as far as we know) elementary particles. This idea is currently being supported by a large number of scattering experiments, we would therefore not expect their mass to be coming from potential energy as nothing binds them together. There have been many alternative ideas to the Higgs mechanism in the literature, for example, a popular alternative was known as technicolor. This is why the discovery of the Higgs boson in 2010 was such an important step in our understanding of particle physics.
Edit What do you mean with 'Rishon theory'? I had to look it up on Wikipedia and I suppose you are referring to a 'possibly new substructure to matter' as is apparently popularized in the recent Star Trek movies? In that case, I think the answer is (as I mentioned above) that there is currently no reason to believe quarks and leptons are not fundamental.