Theoretically. But it's not always easy. Nevertheless, this has been done
for Dinornithidae and Aepyornithidae...
Cooper A., Lalueza-Fox C., Anderson S., Rambaut A., Austin J., and Ward R.
(2001). Complete mitochondrial genome sequences of two extinct moas clarify
ratite evolution. Nature 409: 704-707.
They obtained complete mitochondrial genomes for two moas genera
(_Dinornis_, _Emeus_), and a 1000bp stretch of mitochondrial sequence from
an elephant-bird (_Mullerornis_). I'm guessing that the elephant-bird was a
tough one.