abx has basically answered your question in a comment, but let me add some detail. There is no natural such direct sum decomposition; you are probably thinking of the Bloch-Beilinson *filtration* on Chow groups, whose existence in general is only conjectural. A very nice reference is a paper by Jannsen in one of the "Motives" volumes.

There are some examples where the conjectural filtration is expected to split, so that one really gets a direct sum decomposition. If $X$ is an abelian variety, then there is the Beauville decomposition of the Chow ring, which is defined unconditionally using the Fourier transform. It gives rise to a graduation (as opposed to a filtration) on Chow groups, satisfying all expected properties of the BB filtration except a conjectural vanishing property.

A conjecture of Beauville predicts that for any hyperkähler variety $X$, the Bloch-Beilinson filtration on the Chow groups of $X$ splits. Here a direct sum decomposition satisfying all expected properties was constructed by Beauville and Voisin in the case of a K3 surface.

filtration(the conjectural Bloch-Beilinson filtration). A decomposition exists for some particular varieties, e.g. abelian varieties. $\endgroup$