Let $M$ be a (possibly simply connected) compact manifold $M$. Are there always non-zero classes in the homotopy or homology of $\mathrm{Diff}(M)$ that directly arise from the topology of $M$ itself?

As an example of the type of answers I am looking for I construct non-zero classes in the homotopy and homology of the loop space $\Omega(M)$, which come from the topology of $M$.

Let $M$ be simply connected. Then there is a smallest positive dimension $d$ where $H^d(M)$ is nonzero. Hence $\pi_d(M)\cong H_d(M)$ is non-zero by Hurewicz' Theorem. The long exact sequence in homotopy of the pathspace fibration shows that $\pi_{d-1}(\Omega M)\cong \pi_d(M)$. Applying Hurewicz' Theorem again we see that $H_{d-1}(\Omega M)\cong \pi_{d-1}(\Omega M)\cong \pi_d(M)\cong H_d(M)$. Thus the homology and homotopy have non-trivial elements that come from the topology of $M$.