Vanishing of index of elliptic operators on odd dimensional manifolds

Question

It is known that if $D$ is an elliptic differential operator on $M$ which is assumed to be odd dimensional, then index o $D$ vanishes. It essentially follows from the index formula in cohomology $\langle Ch(D) Td(M),[M] \rangle$ since $Ch(D)Td(M)$ involves only even degree components (powers of chern classes). Is there any way to see this fact (i.e. vanishing of index) directly without using index formula?