We provide numerical evidence in favor of spontaneous chiral symmetry breaking and the concomitant appearance of an Abelian chiral spin liquid for three-component fermions on the triangular lattice described by an SU(3) symmetric Hubbard model with hopping amplitude -t (t > 0) and on-site interaction U. This chiral phase is stabilized in the Mott phase with one particle per site in the presence of a uniform pi flux per plaquette, and in the Mott phase with two particles per site without any flux. Our approach relies on effective spin models derived in the strong-coupling limit in powers of t/U for general SU(N) and arbitrary uniform charge flux per plaquette, which are subsequently studied using exact diagonalizations and variational Monte Carlo simulations for N = 3, as well as exact diagonalizations of the SU(3) Hubbard model on small clusters. Up to third order in t/U, and for the time-reversal symmetric cases (flux 0 or pi), the low-energy description is given by the J-K model with Heisenberg coupling J and real ring exchange K. The phase diagram in the full J-K parameter range contains, apart from three already known, magnetically long-range ordered phases, two previously unreported phases: (i) a lattice nematic phase breaking the lattice rotation symmetry and (ii) a spontaneous time-reversal and parity symmetry breaking Abelian chiral spin liquid. For the Hubbard model, an investigation that includes higher-order itinerancy effects supports the presence of a phase transition inside the insulating region, occurring at (t/U)(c) approximate to 0.07 [(U/t)(c) approximate to 13] between the three-sublattice magnetically ordered phase at small t/U and this Abelian chiral spin liquid.