We introduce the transcendental motive t(X) of a complex cubic fourfold X in P5 and relate the existence of a K3 surface S, such that the (twisted) transcendental motive of S is isomorphic to t(X), with the conjectures about the rationality of X. We also show that the motive of the Fano variety F(X) and of the 8-dimensional hyperk alher variety Z, constructed by Ch. Lehn, M. Lehn, Ch. Sorger and D. van Straten, lie in the same subcategory of the Chow motives generated byt(X) and the Lefschetz motive L.