I'm intrigued by an idea from the World of Alidor blog, which today asks if anyone runs a campaign with no common language. I haven't done that yet, but I might someday, or even soon. The trick to it would be that just because there's no common language in the campaign doesn't mean there doesn't have to be one at the table.
So the DM says, "There's no 'Common' in this game, but you all have to share at least one language so you can communicate with each other. Pick one." In some games, where you automatically start out with some kind of native tongue, and then add languages at-will after that, this would be relatively simple. Everyone just picks a second language to all share. True, this would put limits on certain character concepts; you could only have one "ignorant barbarian who only speaks his native tongue," or at least, only one kind of ignorant barbarian, but in the right kind of game, with flexible characters, that wouldn't be much of a hassle.
For another option, that I think is somewhat more interesting: consider 3e D&D. Each of the races comes with a list of bonus languages; the languages that they're capable of learning, usually for reasons of cultural contact. Certain classes get a few additional bonus languages. Here, finding a common tongue would be a matter of matching the characters' lists against each other in order to find one that everyone shared, and then all taking that language. If there wasn't one that everyone shared, you'd have to find the one that the most characters did, and then negotiate with the DM some kind of background excuse for the remainder to be able to learn that language.
Either way, this system would tell you right at character creation some interesting things about the world and the party. If the language was picked entirely by the players, with no "ignorant barbarian" or "bonus language"-type in-character restrictions, you'd have a very good idea of what they thought was important and who else they wanted to be able to interact with. You've also created a minor reason for the characters to stick together, particularly if they're in hostile territory, or if the language they all speak is unusual for some reason. (This goes double for any group based around all speaking the same language as some barbarian wahoo. Bonus points if the character in question would otherwise be a "lone wolf" type: now you know why Hrathgar the Unruly puts up with this particular bunch of civilized weaklings.)
The "bonus languages" system, on the other hand, tells the players a lot about the world, and how they fit into it. Take a look at the way the 3e languages are set up with the common races. Odds are decent your party is going to end up all speaking Orc, Goblin, or Gnome, which tells you something about how those races fit into the world. A DM could probably get even more mileage out of this if they designed these lists themselves with this purpose in mind. What's a good way to show that a race is everywhere and trades with everyone? Make their language a common bonus language. What's a good way to show that two races don't interact much? Set it up so they don't share a language, and the players have to create some kind of unusual situation to explain why they can talk to each other. Want to make it clear that two races don't trust each other? Write the lists so that they share a language, but it's the language of a mutual enemy. "Of course you can't trust those guys -- the only language they understand is Orc!"
You could even, in a more complex setting, compile lists based on factors beyond race. Because I'm weird, I'm really tempted now to run a game where everyone comes from a different culture but shares a common religious language. Between race, culture, class, and religion, you could end up with a fairly complex set of possible languages. You'd need players who were interested enough in the kinds of things this lets you do to be willing to puzzle out a common language out of all those disparate powers, and willing to tweak their culture or religion in order to get everything to fit together, but I know some players who really enjoy that kind of thing, and others who don't care as long as someone else does the work of figuring it all out.
This doesn't solve the other problem that "common" handles pretty neatly, that of the possibility that one or more party members won't speak the language that most NPCs use. Language confusion can cause a lot of problems at the table, especially if it's a regular thing. But I think that problem is largely resolvable, and I'm going to give some thought over the next day or two to some possible ways to handle it.