Following the answer of Sasha, I can give you here an explicit example of computations using GRR. Suppose you have a smooth projective surface $S$ and a $(-2)$-curve $i : C \hookrightarrow S$, that is $C^2=-2$, $C\simeq \mathbb{P}^1$. Then for any integer $a\in\mathbb{Z}$ the object $i_\ast O_C(a)$ is a spherical object in $D^b(S)$, and thus you can consider the spherical twist (in the sense of Seidel and Thomas) $T:= T_{\mathcal{O}_C(a)}$.
Now the action $T^H$ of $T$ on the cohomology $H^{\ast} (S,\mathbb{Q})$ (following Huybrechts, "Fourier-Mukai Transforms in Algebraic Geometry") is easily computable once you know the chern character $ch(i_*\mathcal{O}_C(a))$. I do these computations in cohomology.:
Write $ch(i_\ast O_C(a)) = (ch_0, ch_1, ch_2)$. By GRR for $i : C \hookrightarrow S$ you have
$$Todd(S) ch(i_\ast O_C(a)) = i_\ast (ch(O_C(a)) Todd(C)) = i_\ast([C]+(a+1)[x])$$ for $[x]$ the (cohomology) class of a point in $C\simeq \mathbb{P}^1$. You can get rid of $Todd(S)$ using that $K_S\cdot C=0$, and you obtain $$ch(i_\ast O_C(a)) = [C]+(a+1)[x].$$