Piecewise isomorphism versus equivalence in Grothendieck ring $\DeclareMathOperator\Var{Var}$Let $K_{0}(\Var_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of a variety, $X$, in $K_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are varieties then we say that they are piecewise isomorphic if there are finite locally closed stratifications, $\{X_{i}\}$ of $X$, and $\{Y_{i}\}$ of $Y$, so that there exist isomorphisms $X_{i}\rightarrow Y_{i}$.
Note, if $X$ and $Y$ are piecewise isomorphic then we have $[\,X\,]=[\,Y\,]$ in $K_{0}(\Var_{\mathbb{C}})$. I expect the converse should be false, perhaps even generically so (in some sense).
Question. What is a simple example of a pair of varieties with equivalent classes in $K_{0}(\Var_{\mathbb{C}})$ which are not piecewise isomorphic?
Here is a candidate example; let $C$ be the affine cone inside $\mathbb{A}^{3}$ given by the equation $x^{2}+y^{2}+z^{2}=0$, and denote by $q\mathrel{:=}[\,\mathbb{A}^{1}\,]$ the Lefschetz motive. Then we have $[\,C\,]=q^{2}$, since removing the cone point we obtain a (Zariski locally trivial) $\mathbb{C}^{*}$ bundle over $\mathbb{P}^{1}$. I expect that $C$ is not piecewise isomorphic to $\mathbb{A}^{2}$. Another example is given by $\operatorname{SL}(2)$, which has class $q^{3}-q$, and which I presume is not piecewise isomorphic to $\mathbb{A}^{3}$ with a line removed.
Edit. As noted in the comments the example $C$ above is in fact piecewise isomorphic to the affine plane.
 A: There are no simple examples as yet; it's been an open question going back to at least Larsen and Lunts - Motivic measures and stable birational geometry, which has been open for about 15 years, and some of us believed that it should be true.
The first counterexample for smooth non-projective varieties was constructed by Borisov as a consequence on his work on L-zero divisors: Borisov - The class of the affine line is a zero divisor in the Grothendieck ring.
There are currently no counterexamples known for smooth projective varieties. Specifically it is not known if $X$, $Y$ are smooth connected projective varieties over a field of characteristic zero such that $[X] = [Y]$, whether $X$ and $Y$ must be birational; they are stably birational by the work of Larsen and Lunts above.
A: After a little digging in the literature, I found the following example:
Theorem. [KS18, Thm. 1.9] There exist non-isomorphic K3 surfaces $X$ and $Y$ over $\mathbf C$ such that
$$[X \times \mathbf A^1] = [Y \times \mathbf A^1].$$
Now if $X \times \mathbf A^1$ and $Y \times \mathbf A^1$ were constructibly isomorphic, then in particular they would be birational. But stably birational surfaces are birational (in this case I believe this follows simply by considering minimal models), which in the case of K3 surfaces would imply $X \cong Y$.

References.
[KS18]  A. Kuznetsov and E. Shinder, Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. Selecta Math. (N.S.) 24.4 (2018), p. 3475–3500. ZBL06941785.
