"FATE grinding" is a pretty fast way of getting it done, especially when you are in an area that have groups already making it happen. FATE pops, more often than not 2 - 5 minutes later you could score 2k - 15k experience depending on your level. Sometimes even more.

However, when it comes time to doing higher level dungeons for story line, getting your relic, earning myth and phil scrolls, higher level dungeons will be really difficult to run through if you are not already experienced with dungeon and pull mechanics for your class. Not to mention Titan (hard mode). I had only been a tank and hated the idea of FATE running, so did dungeons the whole time. You learn the particulars for your class, and it can be fun. Not learning them, dungeons can be a pain in the a**, frustrating and dreaded. Not to mention for the party.

Interspersing FATE grinding (with a lot of dungeon runs like baopee and Darlan_Ashwood above mentioned) would be best, as it gets you prepared for your higher level class when you get there, and the FATEs just makes it faster. But standing there in a FATE doing one thing over and over won't prepare you for dungeons later on, where you will have to truly know your class to get the most out of it. Like I am finding out with my WHM...