Charger: If you're a paladin and completed the epic questline to upgrade your warhorse.
Dreadsteed of Xoroth: If you're a warlock and completed the epic questline to upgrade your mount.
Swift Flight Form: If you're a druid and completed the epic questline to upgrade your normal flight form.
A Tribute to Immortality: This is an intensely difficult raid achievement that I suspect is not going to get substantially easier until well after Cataclysm is released. It's essentially a version of the old Naxx-25 achievement Immortal, and that it's a Feat of Strength says a great deal about the skill and sheer luck that Immortal required. It was only yesterday that anyone got it at all, with Ensidia nabbing the world-first.
Onyxia's Lair (Level 60): "But wait!" you say. "The level 60 Onyxia isn't here anymore!" True, but the achievement is currently linked to the possession of an item dropped by the level 60 Onyxia, and Nefarian now drops the Tier 2 helms that Ony once did. So, yes, it is entirely possible to get the Feat of Strength for killing a level 60 Onyxia by killing a level 60 Nefarian, as long as you get one of the tier helms he hands out. Blizzard is likely to fix this, so if you want the achievement, I'd move on this quickly.
Sulfuras, Hand of Ragnaros: Like its brother, Thunderfury, this remains one of the all-time coolest weapons in the game. Unsurprisingly, it's also one of the biggest pains in the ass to get. Feral Aggression has a good post here on exactly what you can expect to go through trying to get this thing. Bear in mind that this could be considerably harder if you're on a newer realm with a lot of rerolls or newer players, because the odds of finding a Blacksmith with the Sulfuron Hammer recipe are not high in those circumstances.
Thoridal, the Stars' Fury: Easy as pie -- just kill Kil'jaeden and make sure you don't have any rogues in the raid. I kid, I kid. Sunwell is still a pretty tall order, especially if you're pugging, but with a good group, KJ and assorted friends are very much worth your time. Sunwell remains some of the best, most interesting, and most challenging raid content Blizzard's ever done, and nowadays we have the luxury of seeing it without waking up, as I once did, from a sound sleep at night screaming, "Shadowsword Berserker has gained Flurry! Shadowsword Berserker has gained Flurry!"
Go ahead. Ask me what pre-nerf M'uru was like.
Thunderfury, Blessed Blade of the Windseeker: If you can use a 1H-sword, you probably want this. If you can't use a 1H-sword, you want to kill everyone who does, because they're needing on every damn elementium ore drop from Blackwing Lair, aren't they? Thunderfury's Wowhead page gives a good rundown on the (very involved) process you'll need to go through for the game's most famous and iconic sword, but you're not going to get too far until you can coax both Bindings of the Windseeker out of Baron Geddon and Garr (at a 3% drop for each) in Molten Core. If you're a major achievement hound, you're going to be spending a lot of time in MC anyway, so cross your fingers and keep hoping that eventually both of Ragnaros' lieutenants will cooperate.
Val'anyr, Hammer of Ancient Kings: Hard-mode Ulduar-25 is the most straightforward means of getting the 30 Val'anyr fragments you'll need to start the Shattered Fragments questline, and while it's not exactly progression content anymore, it's also no walk in the park. Because each Val'anyr represents a considerable time investment on the part of the raid, this is not a casual, work-at-it-when-you-can thing the way so many of the other Feats of Strength are. Unless you're a paladin, priest, shaman, or druid healer in a competent 25-man raiding guild and your raid leader wants you to have one, do not count on getting Val'anyr for the length for Wrath.
Warglaives of Azzinoth: I don't think anyone on my server managed to get a complete set of Warglaives for the length of BC, which was really a shame because they're easily on the same coolness level held by Thunderfury and Sulfuras. Black Temple is pretty easy for a Wrath-era raid to do, and the most difficult part of the Illidan fight (phase 2) is a shadow of its former terrifying self. People who want to complete their Tier 6 sets and/or who want a crack at Illidan's vanity items (principally felpuppy-on-a-stick, Cursed Vision of Sargeras, and the Bulwark of Azzinoth) will probably be happy to accompany you.
Avast Ye, Admiral!: Completing the Bloodsail grind to Friendly reputation and then turning in the quest Avast Ye, Admiral! will reward you with the Blood Parrot noncombat pet, this Feat, and one headache of a grind back to neutral with the goblin factions.
Friends In High Places: Participating in the Recruit-A-Friend program will reward a Zhevra mount, this Feat, and the opportunity to play with a buddy. Unless, of course, you bought an extra account just to level alts with RAF and got the mount as a bonus. You schemer, you.
Hero of Shattrath: For me, this falls pretty squarely into the same territory trod by people going for their Insane titles -- you have to get exalted with both the Scryers and the Aldor in the BC hub city Shattrath. No, you can't get exalted with both of them at the same time; this requires building reputation with one to exalted, then dropping it and leveling reputation with the other to exalted. How this makes you a hero of Shattrath, I have no idea, because if I were a member of either faction I'd be highly suspicious of your motivation for doing such a thing, but I guess the naaru have their own funky sense of morality. If you're exalted with the Scryers and need to go Aldor, do this quest to get to neutral; if you're exalted with the Aldor and need to go Scryer, do this quest. Both are given by NPC's in Lower City.
Insane in the Membrane: Our own Michael Sacco wrote an OverAchiever on how to do this grind (and ye gods, is it ever a grind), and it's strictly for people with a lot of time on their hands and an obsessive-compulsive desire to get to exalted with as many utterly useless factions as possible. While I haven't done Insane and probably never will -- I'm already exalted with all four Steamwheedle factions, and the thought of re-running Dire Maul to grind back to exalted after grinding to Bloodsail Honored makes me break out in hives -- I can tell you that in the long run this achievement may wind up paying for itself or, at the very least, leaving you less poor than expected. Having to run DM so many times for the goblin factions rewards you with a ton of runecloth and item drops, and I made somewhere in the range of 1,700-2,000 g in a week just grinding the goblin reps from Revered to Exalted. This won't make Insane any easier, mind you, but a fatter wallet will speed things up a bit.
If all else fails, you can snag a Feat of Strength by logging on during WoW's upcoming anniversary, which will reward you with WoW's 5th Anniversary and a noncombat pet. The first day will be November 23rd, 2009, and if it's anything like the 4th anniversary, you should have several days to get it.