Classes of projective homogeneous varieties in K_0 of varieties and of motives

The Grothendieck ring of algebraic varieties, $K_0(\mathrm{Var}_k)$, combines birational geometry with a motivic perspective.
However, many questions concerning classes of projective homogeneous varieties (e.g. Severi--Brauer varieties and quadrics) are more easily solved in $K_0(\mathrm{motives})$ than in $K_0(\mathrm{Var}_k)$.
I will talk about the relations between the classes of motives, when one can lift them to $K_0(\mathrm{Var}_k)$ and why one expects to be able to do that all the time.