In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.