I have once played a game in which every language was, in fact, spoken differently.

We used Dutch, our native language for Common, since that would be the most commonly used language.
Elven was French, Dwarvish was English, and Orcish was German. (We just matched the languages we know with the way we expected the races to talk. The English accent for dwarvish is perfect, and if you speak French with a very soft and slow voice, you're about as close to Elven as you can get. The orcish german was just hilarious, because we seriously overdid it. )

In one case, there was even a short Draconic conversation between a wizard and something else. We had run out of modern languages (the DM didn't speak Spanish, unfortuntely), so we fell back on a few lines of Latin. :)

All in all, the adventure (which ended after 3 or 4 sessions) was great fun, because of:
1. The texture it adds. Those kinds of little details are just nice to have in your game.
2. How much more fluently the game runs. You don't have to say: "Then X speaks in Dwarvish, so no one else can understand him:..." Language transitions are done smooth and efficient.
3. Word jokes. Misinterpretations between two groups were sometimes hilarious to listen to.
"'Armoire'?" said the dwarf, "How do you mean, an arms war? Well, I certainly can't imagine one without them!" :P

Problem is, you and your fellow players do need a pretty good or at least basic knowledge of several languages. And your DM must know all of them. Around here, in Belgium, that's not so hard, because we learned all of the above languages at school anyways. (Well, Latin is of course optional...)