According to Bloch's conjecture, all surfaces of general type $S$ with $H^0(S,\Omega^2_S)=0$ should have $CH(S)$ finitely generated, hence also $K_0(S)$ -- at least modulo torsion. This is probably a very difficult conjecture, but it has been checked in a number of examples : see for instance this preprint and the references therein.

According to Bloch's conjecture, all surfaces of general type $S$ with $H^0(S,\Omega^2_S)=0$ should have $CH(S)$ finitely generated, hence also $K_0(S)$. This is probably a very difficult conjecture, but it has been checked in a number of examples : see for instance this preprint and the references therein.