If you are wearing one of the new faction tabards (e.g. Thunder Bluff, Stormwind, Orgrimmar, Ironforge, etc.), you will only receive reputation gains for that city/faction when in Classic (1-60), Wrath of the Lich King (70-80), and Cataclysm (80-85) dungeons. Outland dungeons will not currently provide reputation gains for the "home city" faction you are championing; this is intended.

Also, just as a heads up, if dungeon mobs and bosses are trivial to you, you will receive less reputation than if they are within your level range. This is just to ensure that players who exceed the dungeon's intended level range can still benefit from championing a specific faction, but aren't given a distinct advantage over those who are of the appropriate level and are unable to solo the content. This should also help encourage players to improve their reputation in dungeons more appropriate for their level, rather than running lower level content solely for the reputation (which isn't a bad thing by any means; we just want you to enjoy the content designed for your level range, too).

(...) To my understanding, both normal and Heroic Wrath of the Lich King dungeons should award reputation to players who are wearing the new faction tabards. There may be additional requirements, however, that aren't overly explicit.

(...) There are technical limitations to overcome with respect to the Outland dungeons. It's something we're interested in pursuing, though, and will be keeping in mind for a future patch.