Pegasus = Greek
Phoenix = Egyptian
Roc = Persian
Harpy = Greek
Romans are believe to have practised falconry and have had eagles on the war banners (Aquila and Reichsadler) so an eagle of some sort makes sense, with the Golden Eagle being most recognisable and correct for the area.
They did also absorb other countries beliefs, especially after conquering them, so Greek mythology could be used even if not 100% fitting.