Qing Liu's example probably works, only we don't know if an abelian variety in
positive characteric is determined by its class in
$K_0(\mathrm{Var}_k)$. However, we do know that in characteristic zero (this is
what Bjorn Poonen uses in his examples) and the non-cancellation is a purely
arithmetic phenomenon and hence can be realised in characteristic zero.

Hence, we let $\mathcal A$ be a maximal order in a definite (i.e., $\mathcal
A\otimes\mathbb R$ is non-split) quaternion algebra over $\mathbb Q$. There is
an abelian variety $A$ over some field $k$ of characteristic zero with $\mathcal
A=\mathrm{End}(A)$ (Bjorn works hard to get his example defined over $\mathbb
Q$, here I make no such claim). For any (right) f.g. projective (i.e., torsion
free) $\mathcal A$-module $M$ we may define an abelian variety
$M\bigotimes_{\mathcal A}A$ characterised by $\mathrm{Hom}(M\bigotimes_{\mathcal
A}A,B)=\mathrm{Hom}_{\mathcal A}(M,\mathrm{Hom}(A,B))$ for all abelian
varieties (concretely it is constructed by realising $M$ is the kernel of an
idempotent of some $\mathcal A^n$ and then taking the kernel of the same
idempotent acting on $A^n$). In any case we see that $M$ and $N$ are isomorphic
precisely when $M\bigotimes_{\mathcal A}A$ is isomorphic to
$N\bigotimes_{\mathcal A}A$.

Now (all the arithmetic results used below can be found in for instance Irving
Reiner: Maximal orders, Academic Press, London-New York), the class group of
$\mathcal A$ is equal to the ray class group of $\mathbb Q$ with respect to the
infinite prime, i.e., the group of fractional ideals of $\mathbb Q$ modulo
ideals with a strictly positive generators. As that is all ideals we find that
the class group is trivial. Furthermore, we have the Eichler stability theorem
which says that projective modules of rank $\geq2$ are determined by their rank and
image in the class group and hence are determined by their rank (the rank
condition comes in in that $\mathrm{M}_k(\mathcal A)$ is a central simple
algebra which is indefinite at the infinite prime). In particular if $M_1$ and
$M_2$ are two rank $1$ modules over $\mathcal A$ and $A_1$ and $A_2$ are the
corresponding abelian varieties we get that $A_1\bigoplus A_2\cong A\bigoplus A$
as the left (resp. right) hand side is associated to $M_1\bigoplus M_2$ (resp.
$\mathcal A^2$). Therefore, to get an example it is enough to give an example of
an $\mathcal A$ for which there exist $M_1\not\cong M_2$. The number (or more
easily the mass) of isomorphism classes of ideals can be computed using mass
formulas and tends to infinity with the discriminant of $\mathcal A$. It is
interesting to note that when the discriminant is a prime $p$ we can go
backwards using supersingular elliptic curves: The mass is equal to the mass of
supersingular elliptic curves in characteristic $p$ and the latter mass can be
computed geometrically to be equal to $(p-1)/24$.