As we talked about a few weeks ago, Cataclysm reputations can provide significantly useful gear. There are 20 epic items available for purchase with reputations, all at item level 359. There are dozens of item level 346 items available.
Picking up these items as you level up your reputations will quickly get you ready for heroics (not to mention, improve your character stats significantly). Since you only need an ilevel 329 average, a few of these epic pieces will make a huge difference to your overall item level.
Each faction's tabards make it very easy to get your reputation up quickly. Pick out the epic gear that you want, and stay focused on that faction until you hit exalted.
Dungeons are cool
Normal difficulty dungeons provide item level 333 gear. It's slightly above what you need to get into heroics, so it will normally take a long time to hit your goal if you raise your item level only through dungeon gear. Still, normal instances are presumed to be the standard for improving your gear until you're in heroics.
While you're in dungeons, make absolutely sure you're wearing the appropriate tabard for your reputation efforts. Just as importantly, consider rolling greed on some drops that you might not use otherwise. For example, if no one else wants a pair of cloth pants that drop, even a paladin might consider rolling on them for the sake of raising his item level.
Of course, you should absolutely be polite about this process. Don't snake good gear from people who will actually use it to improve their characters. That's just not cool.
Finish up Uldum
The gear that's rewarded from Uldum can be as high as item level 325. Sure, that's not enough to get you into heroics; it's just short of the required 329. However, if your extant gear is lower than 325, it can still help get your overall item level a little higher. Instead of burning time standing around, hoping to get into a normal dungeon, you can always keep working Uldum.
Auction house and professions
A lot of item level 359 gear
is created from professions. There are at least a few pieces of epic 359 gear for every class and spec. Like reputations, picking up just a few pieces from the auction house or friends with appropriate professions can make a huge difference to your overall item level.
Also, not every gear replacement has to be straight to 329 or above. Since the required gear level is merely 329, one or two pieces of very sub-par gear could be keeping your item level too low. Even green gear that's closer to 329 could nudge your item level up.
In the end, if you're trying to get into heroics, you just need to keep playing. Work on your reputations, run dungeons, finish up Uldum, and pick up some crafted gear along the way. You will get into heroics fast enough -- just keep trying.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to pulling together enough cash for mid-level expenses such as mounts, to dungeoneering and travel tips for lowbies.