There are no simple examples as yet; it's been an open question going back to at least Larsen and Lunts https://arxiv.org/pdf/math/0110255.pdf, 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: https://arxiv.org/pdf/1412.6194.pdf.
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.