The Hilbert scheme$X^{[3]}$of length-3 subschemes of a smooth projective variety$X$is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow–Künneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map$X^{3}{\dashrightarrow}X^{[3]}$. The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc.240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow–Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety$X$has a multiplicative Chow–Künneth decomposition, then the Chow rings of its powers$X^{n}$have a filtration, which is the expected Bloch–Beilinson filtration, that is split.