K-equivalence $\implies$ isomorphism of (rational) Chow motives is a conjecture going back to this 2002 paper of Wang (Conjecture 2.2); see this overview paper 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. In 2016 the case of semismall K-equivalences has been proved by Liu: "Motivic equivalence under semismall flops".