  Motivated by a recent first principles prediction of an anisotropic cubic
Dirac semi-metal in a real material Tl(TeMo)$_3$, we study the behavior of
electrons tunneling through a potential barrier in such systems. To clearly
investigate effects from different contributions to the Hamiltonian we study
the model in various limits. First, in the limit of a very thin material where
the linearly dispersive $z$-direction is frozen out at zero momentum and the
dispersion in the $x$-$y$ plane is rotationally symmetric. In this limit we
find a Klein tunneling reminiscent of what is observed in single layer graphene
and linearly dispersive Dirac semi-metals. Second, an increase in thickness of
the material leads to the possibility of a non-zero momentum eigenvalue $k_z$
that acts as an effective mass term in the Hamiltonian. We find that these lead
to a suppression of Klein tunneling. Third, the inclusion of an anisotropy
parameter $\lambda\neq 1$ leads to a breaking of rotational invariance.
Furthermore, we observed that for different values of incident angle $\theta$
and anisotropy parameter $\lambda$ the Hamiltonian supports different numbers
of modes propagating to infinity. We display this effect in form of a diagram
that is similar to a phase diagram of a distant detector. Fourth, we consider
coexistence of both anisotropy and non-zero $k_z$ but do not find any effect
that is unique to the interplay between non-zero momentum $k_z$ and anisotropy
parameter $\lambda$. Last, we studied the case of a barrier that was placed in
the linearly dispersive direction and found Klein tunneling $T-1\propto
\theta^6+\mathcal{O}(\theta^8)$ that is enhanced when compared to the Klein
tunneling in linear Dirac semi-metals or graphene where $T-1\propto
\theta^2+\mathcal{O}(\theta^4)$.
