Since I have done some degree of research in this field, I thought I ought to state my conclusions/opinions on the subject of Tyrannosaurus predation:
Among large modern terrestrial hypercarnivores (a useful term mammalogists use for animals which derive >95% of their food in the form of vertebrate flesh), there are three major predatory repetoires:
Comparing theropods to these repetoires, it is fairly easy to relate different groups to the three catagories. Dromaeosaurids are excellent candidates for grapple-and-slash predators, since they proportionately short and stout legs (forget ever reference you've seen to Velociraptor and Deinonychus as being "swift" as dinosaurs go. Even Tyrannosaurus rex has proportionately longer lower legs and feet than do these smaller forms). The claws of the hand and the sickle-claw of the foot match the proportions and angle of felid claws very nicely, and the tail of dromaeosaurids has been known to be a dynamic stabilizer since 1969. And of course, the fighting Velociraptor specimen is in classic grapple-and-slash predatory attack, inculding the disemboweling kicks to the belly of the Protoceratops.
Most large theropods (allosauroids, megalosauroids, Dryptosaurus, etc.), match some variation on the grapple-and-bite theme. The hand claws of these animals closely match the proportions and angles of predatory birds, and are at the end of short but powerful arms. Like predatory birds, these claws were probably not the primary weapons of killing, but were used to seize and hold the prey while the jaws did the work. Note that it is these animals, and not dromaeosaurids, which match modern "raptors" the best.
Tyrannosaurids fit well with the pursuit-and-bite catagory. Like canids and hyaenids, they have proportionately long legs (T. rex itself has legs which are more "cursorial" than the man-sized herbivore Dryosaurus and other accepted runners), very powerful jaws, and claws of the hand and feet which are not highly curved and rounded in cross-section. Although they may not have pursued prey for wolf-like distances, the body of anatomical evidence points to the adaptations of tyrannosaurids as being predatory, and specifically pursuit-and-bite predatory, features.
And as for scavenging - none of the alledged scavenging features suggested by Horner holds up in quantitative or comparative analysis. His claim that predators need to use their forelimbs in prey acquisition does not stand the test of observations of the modern world. Tyrannosaurids show more cursorial adaptations than any other large Late Cretaceous Asiamerican dinosaur (hadrosaurids, ceratopsids, ankylosaurids, etc.), so they probably were faster than any of these.
BUT... as others have already pointed out, scavenging and predation are not mutually exculsive behaviors. In some regions of Africa, for example, lions are predominantly scavengers and hyaenas the major predators, while in other parts of the same continent, these roles are reversed. Tyrannosaurids would be in a good position to bully any other theropod away from a corpse (dromaeosaurids arguably may be more deadly pound for pound, but tyrannosaurids had a LOT more pounds...). It is not unreasonable that certain individual tyrannosaurid populations, or even species, may have gotten most of their food from carcasses. Nevertheless, the anatomy of tyrannosaurids indicates that they were capable of dispatching prey using techiniques grossly similar to those used by canids, hyaenids, and the like - running down animals, seizing them in their jaws, and ripping out huge chunks and/or suffocating the prey item until it was dead.