Here is how the heat kernel proof of Atiyah-Singer goes at a high level. Let $(\partial_t - \Delta)u = 0$ and define the heat kernel (HK) or Green function via $\exp(-t\Delta):u(0,\cdot) \rightarrow u(t,\cdot)$. The HK derives from the solution of the heat equation on the circle:
$u(t,\theta) = \sum_n a_n(t) \exp(in\theta) \implies a_n(t) = a_n(0)\cdot \exp(-tn^2)$
For a sufficiently nice case the solution of the heat equation is $u(t,\cdot) = \exp(-t\Delta) * u(0,\cdot)$.
The hard part is building the HK: we have to compute the eigenstuff of $\Delta$ (this is the Hodge theorem). But once we do that, a miracle occurs and we get the
Atiyah-Singer Theorem: The supertrace of the HK on forms is constant: viz.
$Tr_s \exp(-t\Delta) = \sum_k (-1)^k Tr \exp(-t\Delta^k) = const$
For $t$ large, this can be evaluated topologically; for small $t$, it can be evaluated analytically as an integral of a characteristic class.
Edit per Qiaochu's clarification
Supposedly (I don't have ready access to this) an
This article of Kotake (really in here as the books seem to be mixed up) proves Riemann-Roch directly à la McKean and Singer. The book of Rosenberg I mentioned in the comments illustrates using the key points of McKean-Singer. See also hereheat kernel.