From what I have read, it appears that rishons are supposed to be constituents of quarks, in theory. This alone does not conflict with the structure of mesons or baryons. As far as grouping sub-atomic particles into groups of three, this does already occur in some cases. The pions are a group of three similar particles; we call them an "isospin triplet". There are isospin triplets in every multiplet of mesons and baryons, as the multiplets for mesons in SU(3) include nine members, and the multiplets for baryons may include either eight, nine, or ten members. All mesons are arranged into groups of nine (nonet) that break down into one group of three (triplet), two groups of two (doublets), and two individuals (singlets). The baryons get a little more complicated.
I do not see how rishons generate any conflict in this regard, since the grouping of three they refer to is the (u, c, t) and (d, s, b) grouping of quark types by charge. This has no specific bearing on mesons, which are constituents of a quark and an anti-quark of any flavor/type.
In my personal opinion, I find the theory on rishons to be quite unnecessary and underdeveloped. It makes things more complicated than they need to be, and still leaves unresolved issues of the same type that it attempts to resolve in the first place (especially by trying to explain the 2/3 versus -1/3 charges of quarks by "masculine" and "feminine" rishons that have different fractional charges themselves!). Not only that, but we still have no reason to believe that quarks are not point-like as they are, so I wouldn't put any stock in rishons just yet.
