Bott vanishing for Fano 3-folds

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that $H^j(X,\Omega^i_X\otimes L)=0$ for every $j>0$, $i\geq 0$, and $L$ ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are many more than expected. Along the way, we conjecture that for every projective birational morphism $\pi\colon X\to Y$ of smooth varieties, and every line bundle $A$ on $X$ that is ample over $Y$, the higher direct image sheaf $R^j\pi_*(\Omega^i_X\otimes A)$ is zero for every $j>0$ and $i\geq 0$.