Evolutionary biologist Olivia Judson imagines that they may have.
As she writes, there are scant signals cluing us in to the sexual life of the male T. Rex. Did he have a penis? Or perhaps just a cloaca, which avians, amphibians, and reptiles use to excrete waste, semen, or eggs (should he actually be a she) and secrete attractive scents from a little gland tucked away inside the cloaca.
While male birds lack intromittent organs,* reptiles such as crocodiles have a penis hiding inside their cloaca. It's so well-hidden, in fact, that it's difficult to identify the sex of a crocodile without getting very, well, personal with the male - the clitoral growth of female crocodiles can rival that of a male's penis, and one can easily be mistaken for the other.
It's interesting to note that crocodiles' sex is determined by the temperature at which the egg is incubated - eggs incubated at low (23-26 C) or high (30-33 C) temperatures are more likely to produce female offspring while those incubated at intermediate temperatures (27-30 C) produce both males and females (see here for more). This pattern, known as temperature-dependent sex determination (TDSD), could very well apply to our friend T. Rex as well.
And intriguingly, researchers very recently revealed that TDSD doesn't just influence who you are, but how you perform. Incubating eggs at their optimal sex-determining temperature produces offspring that are maximally fit for reproduction - male lizards incubated at just the right male-determining temperature were more successful mates.
But, as Judson reminds us, we may never know. We can only glean small clues from fossil records, which haven't included a penis bone or baculum. Which, sadly, might have told us a bit more about T. Rex's reproductive prowess (or at least his success as a mate), as baculum length correlates with testis mass in some species.
So I suppose that we'll just have to wonder along with Judson...
Funny Semi-Related Research Link: how ostrich semen is collected...using the avian version of a blow-up doll
* Not penises; thanks to Smart Reader for clarification, below
male birds lack penises
Allow me to be the first to disabuse you of that misconception.
(though in fairness, most birds do, in fact, lack an intromittent organ)
Correct - birds do not have penises. Incorrect - some birds do have intromittent organs (or phalluses/phalli). And the big question is why some of those get to be quite long.
Based on the fact that crocodilians and basal birds both have a phallus (or whatever you want to call it*), tyrannosaurs and other Mesozoic dinosaurs surely did too... and in fact it's pretty difficult to imagine how they reproduced without one. We have someone studying this at University of Portsmouth (UK), and one of the most amusing parts of the project has been the effort to realistically model mating postures for the animals. With a tyrannosaur it's not that difficult - stegosaurs are another matter of course.
* Not everyone agrees that we should limit the term penis to mammals. See my turtle phallus post here.
Ahh, this is exactly the kind of spirited discussion that got me into biology in the first place. No holds barred free-for-alls! Sometimes going into the wee hours of the night and even spilling out into the streets. We loved biology then.