  We show that a wide range of Fano varieties of K3 type, recently constructed
by Bernardara, Fatighenti, Manivel and Tanturri, have a multiplicative
Chow-K\"unneth decomposition, in the sense of Shen-Vial. It follows that the
Chow ring of these Fano varieties behaves like that of K3 surfaces. As a side
result, we obtain some criteria for the Franchetta property of blown-up
projective varieties.
