In my current D&D campaign, we use the Realms, so there's no unusual crosbreeding going on.
I have some ideas for a homebrew setting (where elves are the central race, dwarves are elementals/outsiders, and humans a minor race of savages - I don't like that most campaigns follow the same storyline: in the past, there were great elven empires, but now humans run the show. And since we have elf-less campaigns, I think it's time for a campaign where the elves didn't wane) and while I haven't thought too much about it, I guess I'd just leave out half-breeds.
If you want an interestinc half-breed concept, go read up on Midnight:
There, the big races (sans humans) are all decended from the elder fey. Since they have the same origins, they can mate amongst each other, but not with humans (or at least, they're not interfertile). No half-human/half-elves, no half-human/half-orcs.
There are three crossbreeds, though:
Elflings: Elves and halflings are always considered "cousins" since they're generally more similar to each other than to the other fey. And with the jungle elves being smaller even than normal elves, interbreeding between jungle elves and halflings occurs, with elflings being the result. They're extremely agile, but share their parents' frailty and weakness.
Dwarrows: Gnomes and Dwarves Are also somewhat similar, at least physically (generally, the taciturn Mountain Fey have not that much in common with the marcantile and socially adept River Fey), so crossbreeds between those two are numerous enough to give them a name: Dwarrows. They share some characteristics of each parent race, how much of what depending on where they were raised.
Dworgs. It is rumored that orcs were once dwarves who have been corrupted by the shadow. The dwarves of course refuse to even consider any kinship with their hated enemies, but the fact that the two races can crossbreed seems to lend the argument some weight. No race is physically more impressive than dworgs, no race is more pitiable than these results of war crimes against dwarves, and no race harbours more hatred towards the orcs - not even the dwarves (and that counts for a lot).