Half-breeds have been a part of D&D since the addition of half-elves in the 1976 Greyhawk supplement, which were probably inspired by the half-elven offspring of Luthien and Beren from Lord of the Rings. Half-breeds, however, raise some interesting and potentially awkward questions about the nature of a species. Even Tolkien tread carefully around this issue, and considered that elves needed to give up their immortality and live a mortal life if they were to marry a mortal. This was considered such a rare occurrence that there were only three such unions in all the ages. Tolkien further constrained the few half-elves resulting from the union of Luthien and Beren by forcing them to irrevocably choose which kindred to belong to. Elrond chose to be of Elvenkind, while his brother, Elros, chose to be of Mankind. Elros went on to found the kingdom of Numenor and give rise to the lineage from which Aragorn descended (so, yes, Aragorn married his cousin).
The problem with the half-breed is that the Biological Species Concept defines a species as a group of organisms that can interbreed, or potentially interbreed, and produce fertile offspring. Thus, from 1976 on, the implication has been that in D&D humans and elves are the same species. The situation became even more confused with the introduction of half-orcs, thus extending the concept of Homo sapiens to include not only humans and elves, but now orcs as well. The assumption made in D&D is that half breeds are the result of human and elven or human and orcish parentage, but since the species lines have been erased, it is just as likely to have half-breed offspring of orcish and elvish parentage.
While Tolkien handled the matter deftly, making interbreeding between the two species a divine gift of the Valar and a rare occurrence, 3E threw logic out the window with the proliferation of templates allowing one to create dire-fiendish-undead-half-mindflayers if you wanted (assuming you had the patience to work out the five-page-long stat block to accompany it).
So what? After all, Gary Gygax often reminded us that D&D was a game and we shouldn't get too hung up on realism. The thing is though, Gary also always kept one foot firmly planted in reality and was careful to balance good game mechanics with enough realism to create that all-important aspect of role-playing: suspension of disbelief.
For me suspension of disbelief is impossible without verisimilitude, which is achieved by imposing internally consistent natural laws that govern my campaign world. Those laws don't necessarily have to be those of the real world, but there ought to be some structure for how things work. It is perfectly acceptable to have races like half-elves and half-orcs, but I find it far more satisfying to consider the implications of such races and come up with a logical rationale for them.
Say we like both the half-elf and half-orc races and want to keep them. There are a few logical options to ponder:
A) Accept that all three races are, indeed the same species, and develop the world's history based on that assumption - what is their evolutionary history; how and when did they begin to develop such divergent racial traits? Furthermore, what geographical or morphological barriers are in place to restrict the proliferation of half-breeds and the otherwise inevitable convergence back into one homogenous race.
B) Imagine that humans, elves, and orcs are separate, but closely related species who can interbreed to produce infertile offspring, much like domestic horses, Equus ferus can interbreed with donkeys, Equus africanus, to produce infertile mules.
C) Humans, elves, and orcs are not closely related species and mating cannot produce offspring.
This is not to say that it is necessary to rationalize everything, but I find that at least considering the rationale behind elements of the milieu can open up some new perspectives that can help to shake of old cliches and result in a more original and dynamic campaign.
I created my own world of Faeridor about twenty years ago, long before I started thinking of such things and consequently I have had to retro-actively rationalize decisions that I sometimes regret. Half-elves are long-established in the campaign; the Mac Morne bloodline of Llanvirnesse is mingled with that of Sylvani royalty and the fertility of half-elves is unquestionable. Here, the rationale becomes sticky, because while the Sylvani are indigenous to Faeridor, humanity originated on Earth, with various tribes later transplanted to Faeridor by way of a dimensional rift. Thus, it would seem unlikely that two races from different dimensions would be genetically similar enough to produce viable offspring. The answer lies with the gods of man, the Tuatha de Danann who dwelt on Faeridor but dallied also on Earth. If humans and elves are both the by-product of the de Danann, then cross-breeding between the two races becomes better justified as they would share common ancestry.
Orcs, on the other hand, fulfill only a minor role in my world. They, too, are aliens brought to Faeridor by means of a dimensional rift, though in nowhere near the numbers that humans were. They serve as mercenary troops for any warlord who will give them coin and the chance to fight and are present in too few numbers to be a major menace. Consequently, the question of half-orcs has never arisen so it is easy enough to say that elves and humans cannot interbreed with orcs.
This is an inelegant explanation - the result of coming up with it after the fact, but I still find it more satisfying to know how the races came to be and how they relate to each other than simply to shoe-horn them in without considering the implications.
Welcome Back to the Labyrinth
"We have been away far too long, my friends," Ashoka declared, his face lit by the eldritch green glow of his staff. "But we have finally returned to the labyrinth whence our adventures first began."
"Just imagine the treasures that lie within," said Yun Tai, flexing his mighty muscles. "Wealth enough to live in luxury the rest of our days."
"And arcane artifacts of great power," added Ashoka his words dripping with avarice. "All ours for the taking!"
"Umm...guys?" Nysa interrupted. "Do you hear something dripping?"