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.