  We present a family of new solutions to the tetrahedron equation of the form
$RLLL=LLLR$, where $L$ operator may be regarded as a quantized six-vertex model
whose Boltzmann weights are specific representations of the $q$-oscillator or
$q$-Weyl algebras. When the three $L$'s are associated with the $q$-oscillator
algebra, $R$ coincides with the known intertwiner of the quantized coordinate
ring $A_q(sl_3)$. On the other hand, $L$'s based on the $q$-Weyl algebra lead
to new $R$'s whose elements are either factorized or expressed as a terminating
$q$-hypergeometric type series.
