  Highly-elongated quasi-one-dimensional cold atom samples have been studied
extensively over the past years experimentally and theoretically. This work
determines the energy spectrum of two identical fermions and a third
distinguishable particle as functions of the mass ratio $\kappa$ and the
free-space $s$-wave scattering length $a_{3\text{D}}$ between the identical
fermions and the distinguishable third particle in a cylindrically symmetric
waveguide whose symmetry axis is chosen to be along the $z$-axis. We focus on
the regime where the mass of the identical fermions is equal to or larger than
that of the third distinguishable particle. Our theoretical framework accounts
explicitly for the motion along the transverse confinement direction. In the
regime where excitations in the transverse direction are absent (i.e., for
states with projection quantum number $M_{\text{rel}}=0$), we determine the
binding energies for states with odd parity in $z$. These full
three-dimensional energies deviate significantly from those obtained within a
strictly one-dimensional framework when the $s$-wave scattering length is of
the order of or smaller than the oscillator length in the confinement
direction. If transverse excitations are present, we predict the existence of a
new class of universal three-body bound states with $|M_{\text{rel}}|=1$ and
positive parity in $z$. These bound states arise on the positive $s$-wave
scattering length side if the mass ratio $\kappa$ is sufficiently large.
Implications of our results for ongoing cold atom experiments are discussed.
