  We have developed a complete semiclassical Wentzel-Kramers-Brillouin (WKB)
theory for $\alpha-\mathcal{T}_3$ model which describes a wide class of
existing pseudospin-1 Dirac cone materials. By expanding the sought wave
functions in a series over the powers of Planck constant $\hbar$, we have
obtained the leading order expansion term which is the key quantity required
for calculating the electronic and transport properties of a semiclassical
electron in $\alpha-\mathcal{T}_3$. We have derived the transport equations
connecting each two consecutive orders of the wave function expansion and
solved them to obtained the first order WKB wavefunction. We have also
discussed the applicability of the obtained approximation and how these results
could be used to investigate various tunneling and transport properties of
$\alpha-\mathcal{T}_3$ materials with non-trivial potential profiles. Our
results could be also helpful for constructing electronics devices and
transistors based on innovative flat-band Dirac materials.
