ISO an elegant derivation of the Geodesic Equation from the Einstein Equation

The usual way is to assume a non-interacting "dust" energy momentum $T^{munu}= rho u^mu u^nu$ with $nabla_mu( rho u^mu)=0$. Then $$ nabla_mu T^{munu} =0quad Rightarrow quad u^mu nabla_mu u^nu=0 $$ so the integral curves $x(tau)$ (worldlines) $$ frac {dx^mu}{dtau}= u^mu $$ of the particles are autoparallels and therefore geodesics.