Are you considering complete intersections in projective space? In that case, for a complete intersection $X$ in $\mathbb{P}^n$ of type $(d_1,\dots,d_c)$, then $\omega_X$ is isomorphic to $$\mathcal{O}_{\mathbb{P}^n}(-n-1+d_1 + \dots + d_c)$$. Thus $h^0(X,\omega_X)$ is nonzero unless $d_1+\dots+d_c \leq n$. In this case, $X$ is a Fano manifold. Thus, by work of Kollár-Miyaoka-Mori and Campana, $X$ is rationally chain connected (I think you can usually prove this directly for complete intersections; though definitely not for general Fano manifolds). Thus $\text{CH}_0(X)$ is just $\mathbb{Z}$ since every pair of points is connected by a chain of rational curves in $X$.