  We investigate one dimensional attractive Fermi gases in spin-dependent
optical lattices. We show that three-body bound states - "trimers" - exist as
soon as the two tunneling rates are different. We calculate the binding energy
and the effective mass of a single trimer. We then show numerically that for
finite and commensurate densities $n_\uparrow=n_\downarrow/2$ an energy gap
appears, implying that the gas is a one-component Luttinger liquid of trimers
with suppressed superfluid ordering. The boundaries of this novel phase are
given. We discuss experimental situations to test our predictions.
