With heirlooms, expect to gain roughly 1/4-1/2 a level per dungeon, more depending on how thorough your group is with clearing and other variables. Figure in running maybe 2 dungeons an hour if you stick to the shorter ones (and the LFD system breaks up longer ones into sections) and you can grind out 1-68 in less than 5 days played (at which point I personally recommend Northrend questing with dungeons between quest grinds when you get bored).
For those leveling as a tank or healer, LFD is lightning fast and much less boring than solo questing to 80.