  We exhibit several families of Fano threefolds with a multiplicative
Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, a
certain tautological subring of the Chow ring of powers of these threefolds
injects into cohomology. As a by-product of the argument, we observe that
double covers of projective spaces admit a multiplicative Chow-K\"unneth
decomposition.
