  We present a first-principles study of the cross-plane thermal conductivity
$\kappa_{\perp}$ in a wide variety of semiconductor thin films. We introduce a
simple suppression model that matches variance-reduced Monte Carlo simulations
with ab-initio phonon dispersions and scattering rates within $\leq 5\%$ even
for anisotropic compounds. This, in turn, enables accurate $\kappa_{\perp}$
reconstruction from tabulated cumulative conductivity curves
$\kappa_{\Sigma}(\Lambda_{\perp})$. We furthermore reveal, and explain, a
distinct quasiballistic regime characterised by a fractional thickness
dependence $\kappa_{\perp} \sim L^{2-\alpha}$ in alloys (where $\alpha$ is the
L\'evy exponent) and logarithmic dependence $\kappa_{\perp} \sim \ln(L)$ in
single crystals. These observations culminate in the formulation of two compact
parametric forms for $\kappa_{\perp}(L)$ that can fit the first-principles
curves across the entire ballistic-diffusive range within a few percent for all
investigated compounds.
