  Enriques varieties have been defined as higher-dimensional generalizations of
Enriques surfaces. Bloch's conjecture implies that Enriques varieties should
have trivial Chow group of zero-cycles. We prove this is the case for all known
examples of irreducible Enriques varieties of index larger than $2$. The proof
is based on results concerning the Chow motive of generalized Kummer varieties.
