  A three-parameter (positive odd integer $s$, thickness factor $\lambda$, and
asymmetry factor $a$) family of asymmetric thick brane solutions in five
dimensions were constructed from a two-parameter ($s$ and $\lambda$) family of
symmetric ones in [R. Guerrero, R.O. Rodriguez, and R. Torrealba, Phys. Rev. D
\textbf{72}, 124012 (2005).]. The values $s=1$ and $s\geq3$ correspond to
single branes and double branes, respectively. These branes have very rich
inner structure. In this paper, by presenting the mass-independent potentials
of Kaluza--Klein (KK) modes in the corresponding Schr\"{o}dinger equations, we
investigate the localization and mass spectra of fermions on the symmetric and
asymmetric thick branes in an AdS background. In order to analyze the effect of
gravity-fermion interaction (i.e., the effect of the inner structure of the
branes) and scalar-fermion interaction to the spectrum of fermion KK modes, we
consider three kinds of typical kink-fermion couplings. The spectra of left
chiral fermions for these couplings are consisted of a bound zero mode and a
series of gapless continuous massive KK modes, some discrete bound KK modes
including zero mode (exist mass gaps) and a series of continuous massive KK
modes, infinite discrete bound KK modes, respectively. The structures of the
spectra are investigated in detail.
