**K-equivalence $\implies$ isomorphism of (rational) Chow motives** is a conjecture going back to [this 2002 paper of Wang][1] (Conjecture 2.2); see [this overview paper][2] of his for what else K-equivalence should imply on the cohomological level. 

Some important special cases, such as ordinary flops and Mukai flops are proven in [this 2010 paper by Lee, Lin and Wang][3]. In 2016 the case of semismall K-equivalences has been proved by [Liu: "Motivic equivalence under semismall flops"][4].


  [1]: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-60/issue-2/Cohomology-Theory-in-Birational-Geometry/10.4310/jdg/1090351105.full
  [2]: https://arxiv.org/pdf/math/0204160.pdf
  [3]: https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n1-p05-s.pdf
  [4]: https://arxiv.org/pdf/1603.06152.pdf