  We prove that Kodaira vanishing holds for an arbitrary smooth Fano threefold
in positive characteristic. To this end, we show that it is quasi-F-split when
the Picard number or the Fano index is larger than one. We also establish
Akizuki-Nakano vanishing for smooth Fano threefolds when the Picard number or
the Fano index is larger than one, and therefore they lift to the ring of Witt
vectors.
