I've got a number of ideas about ways to deal with the language problem, most of which aren't exclusive; in fact, I think a lot of these would work better if you used them together in some kind of combination. Most of them have to do with campaign design and group interaction, rather than in-game and setting issues, so they're going to be of limited utility in a lot of cases. (I tend to do most of my real campaign building while the players are making their characters, because I'm lazy and procrastinate.) I'm not entirely sure what'll happen with most of these if they're actually tried at the table, so I'd love to hear about it if you do try one, or come up with something along similar lines.
- Everyone speaks elvish? Great! We'll start in the Elven Homelands. Basically, just take the language that the PCs all speak and find a place in the setting where that language is, effectively, "common." This does require that you to be familiar with a fair amount of your setting at the start of the game; it'd be a particularly good option if you're running a published setting, especially if you're not sure where in the setting that you want to set the game, since the players basically pick for you. It also doesn't help much if the characters end up wandering far afield from their starting location, but for some campaign concepts that won't be an issue.
- We only buy things from Gnomes. If the "common language" that all the PCs speak is a trade language of some kind, you can solve a lot of the problem just by running the kind of game where the characters mostly talk to merchants; put most of the action in the dungeon, say, and only require them to deal with aboveground NPCs when they need to sell loot or buy equipment. Then, as long as you have the right kind of players, it doesn't matter what the main local language is, as long as there's a gnome caravan (or whatever) around.
- The Foreign Quarter. If the language that everyone shares is one that's only spoken widely in places that would be sort of weird to set a campaign in (goblin, for instance), then you might set up the campaign so that the dominant language is something else, but the PCs for the most part don't deal with the dominant culture. Instead, they interact mainly with some subcommunity that shares their language; the foreign quarter, or the goblin underclass that runs the city's utilities (and criminal underworld). Ideally, I think, you'd pair this option with rules that made learning new languages possible, if not exactly easy; when the PCs did start to have to deal with the area's upper class on a regular basis, they could then pick up the necessary languages.
- Blue-booking. If the campaign is set up in such a way that most of the vital, everyday interactions use the language everyone shares, but it's still possible to talk to NPCs who not everyone understands, you can handle those interactions outside of the regular game time with the players who are interested.
- Online play. Online play of various kinds can make blue-booking easier; in some kinds of PBP games, it's practically not even a separate activity from the main game. Chat games also have the advantage that players can "get up from the table" without distracting the rest of the group. If the group decides to have a short chat with someone who one or two people can't talk to, the people who aren't involved in that conversation can do something else for a bit. If this started to happen a lot in a given game, I'd look for another solution, but depending on the players, as long as it stays an occasional, temporary event, in chat this isn't much of an issue.
- Babelfish. In the Doom & Tea Parties game, my character interacts with a number of characters on a regular basis who don't all speak the same language as her, but we're getting around it with some semi-permanent magic and a bit of DM handwave. Giving the PCs access to some kind of universal translation magic or technology isn't always the most interesting way to deal with this issue, but it's very simple and effective, and a good way to go if your main problem with a common language is that it wouldn't make any sense for world-building purposes.
- Flavors of Speak Language. On the other hand, universal translation nullifies the possibilities of language as a logistical issue. That's not something that you're never going to want to deal with on a regular basis, but the right group might find it fun to have to solve "the language problem" when they travel to a new place. One option there is to make speak language, or something like it, either permanent or long-term but also very specific. You do have to keep track of who speaks what language, but that information is mainly necessary so you can find the right spell.
Overall, I think "common" is definitely the best way to go if you're just not all that interested in language, which is going to be true of a lot of groups. On the other hand, if you do think language is fun to play with at the table, some of these options might help you make it gameably interesting and avoid the usual frustrations.