  We consider the problem of three distinguishable fermions confined to a
quasi-two-dimensional (quasi-2D) geometry, where there is a strong harmonic
potential in one direction. We go beyond previous theoretical work and
investigate the three-body bound states (trimers) for the case where the
two-body short-range interactions between fermions are unequal. Using the
scattering parameters from experiments on ultracold $^{6}$Li atoms, we
calculate the trimer spectrum throughout the crossover from two to three
dimensions. We find that the deepest Efimov trimer in the $^{6}$Li system is
unaffected by realistic quasi-2D confinements, while the first excited trimer
smoothly evolves from a 3D-like Efimov trimer to an extended 2D-like trimer as
the attractive interactions are decreased. We furthermore compute the excited
trimer wave function and quantify the stability of the trimer with respect to
three-body recombination by determining the probability that three fermions
approach each other at short distances. Our results indicate that the lifetime
of the trimer can be enhanced by at least an order of magnitude in the quasi-2D
geometry, thus opening the door to realizing long-lived trimers in
three-component Fermi gases.
