I've given that question some thought, and this is the theories I've come up with:
1.) As opposed to Rayman 2, Rayman 3 has only one scene which involves going underwater. Now game developers usually have a tight schedule; maybe more important things were left to be done, and they just didn't want to bother fix this for one single Part, or forgot about it in the end.
2.) Sort of what Adsolution said: In Rayman 1, Rayman learned that swimming might be a useful ability, so he learned it, and was able to do it in Rayman 2. In Rayman 2 he learned that he should improve his ability to hold his breath. So now in Rayman 3, Rayman's able to hold is breath for a very long time.
3.) Technically speaking, Rayman shouldn't be able to breathe either way, since (assuming that Rayman's body is similar to that of a human) his lungs and his mouth are in no way connected. Maybe he just doesn't have to breathe, although that's somewhat unlikely taking the previous Rayman games into consideration.
4.) Rayman keeps getting teleported from one world to another. Maybe the Land of the Livid Dead is a world with no water, but a fluid which just looks like water. Maybe it's a fluid which contains air and allows Rayman to breathe just fine, or whatever.
I personally think that it's the first one; most likely the developers just forgot about it.
