  Let $Y$ be a smooth complete intersection of three quadrics, and assume the
dimension of $Y$ is even. We show that $Y$ has a multiplicative Chow-K\"unneth
decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of
(powers of) $Y$ displays K3-like behaviour. As a by-product of the argument, we
also establish a multiplicative Chow-K\"unneth decomposition for double planes.
