The extant dinosaurs, or birds as they are popularly called, form the clade Neornithes (a Greek compound which means "new birds"), corresponding to the classical class Aves (a Latin word which means "birds"). To the best of current knowledge, Neornithes is divided into two lineages:
The Palaeognathae ("old jaws" in Greek): ostriches, emus, cassowaries, and kiwis. There are no palaeognath birds of prey; most palaeognath birds are flighless, and those few which can fly are very poor fliers.
The Neognathae, the vast majority of the birds; the neognath birds are divided into two lineages:
The Galloanserae (Latin for chickens-geese), or fowl; no birds of prey, although some of them are very good fliers.
The Neoaves (a Greek-Latin compound meaning "new birds"). Within the bewildering diversity of the Neoaves there are many lineages, three of which are birds of prey:
The closest living relations of the falcons are the parrots, the sparrows and the song birds. The owls are not far from the eagles, vultures and hawks; both more closely related to woodpeckers than with falcons.
What is important is to notice that all birds of prey conform to the stereotypical image of a bird as illustrated by a sparrow and show obvious similarities: they have "very good eyesight for finding food, strong feet for holding food, and a strong curved beak for tearing flesh" (from Wikipedia). In what degree these are inherited from the common ancestor of the land birds or represent convergent evolution is not yet known.
Since we already have three distinct lineages of birds of prey, we can deduce that this lifestyle provides a comfortable ecological niche. In a world with no Accipitriformes something would have taken their place; either the descendants of the Strigiformes, or the descendants of the Falconiformes, or even the descendants of some other group: but for sure the food would not be left uneaten; the prey would not be left unhunted.
There would still be falcons and hawks and eagles: only scientists will find out that they come from different lineages. After all, eagles and hawks had been classified in the same order Falconiformes as falcons for almost three centuries -- it was only recently that genetic studies showed that they were not closely related.