I understand that the Tannakian formalism (which I only "know" extremely superficially) is very important for the theory of motives. I guess "the" conjectural category of motives should be Tannakian, or a full subcategory of a Tannakian category (I hope what I write makes sense).

Now let $\mathcal{V}_k$ be the category of $k$-varieties with $k$ a field, and consider the Grothendieck ring $K_0(\mathcal{V}_k)$. It is the quotient of the free abelian group generated by the isomorphism classes $[X]$ of $k$-varieties by the relations $[A] = [A \setminus C] + [C]$, where $C$ is a closed $k$-subvariety in the $k$-variety $A$; the fiber product over $k$ then induces a ring structure.

As you may know, some aspects of motives over $k$ can be seen in $K_0(\mathcal{V}_k)$, but many others can not.

My question is: how much does $K_0(\mathcal{V}_k)$ "capture" from the (full subcategory of the) *Tannakian category* which the category of $k$-motives should be ?