We adapt for algebraically closed fields k of characteristic greater than 2 two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface X of dimension 3 over ℂ, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with ℤ2-coefficients) of the minimal class θ4/4! of the intermediate jacobian J(X) of X implies the Chow-theoretic decomposition of the diagonal of X. Using the second result, the Tate conjecture for divisors on surfaces defined over finite fields predicts, via a theorem of Schoen, that every smooth cubic hypersurface of dimension 3 over the algebraic closure of a finite field of characteristic >2 admits a Chow-theoretic decomposition of the diagonal.