  We calculate the lattice thermal conductivity ($\kappa$) for cubic
(zinc-blende) and hexagonal (wurtzite) phases for 8 semiconductors using
$\textit{ab initio}$ calculations and solving the Phonon Boltzmann Transport
Equation, explaining the different behavior of the ratio $\kappa_{\rm
hex}/\kappa_{\rm cub}$ between the two phases. We show that this behavior
depends on the relative importance of two antagonistic factors: anharmonicity,
which we find to be always higher in the cubic phase; and the accessible phase
space, which is higher for the less symmetric hexagonal phase. Based on that,
we develop a method that predicts the most conducting phase---cubic or
hexagonal---where other more heuristic approaches fail. We also present results
for nanowires made of the same materials, showing the possibility to tune
$\kappa_{\rm hex}/\kappa_{\rm cub}$ over a wide range by modifying their
diameter, thus making them attractive materials for complex phononic and
thermoelectric applications/systems.
