Most flute players I've met could stand to rethink the way their bodies interact with the instrument. The flute really is one of the least comfortable instruments to play. It's very easy to develop a great deal of tension in the hands, wrists, and shoulders, which can lead to repetitive strain injury. Let alone that you've simply got to find a way of keeping the thing steady while you're moving your fingers around! Small wonder that there are so many strange postures among traditional flute players. One of the most common is to brace the flute against the left shoulder, or rest it on top of the shoulder--obviously a technique developed by evil chiropractors looking to increase their business.
An ideal flute posture is one that allows you to (a) stand or sit up straight, (b) maintain a comfortable, relaxed, secure hold on the instrument, (c) have your fingers totally free from having to support the flute, (d) play with a correct embouchure, and (e) make maximum use of your lung capacity, all (f) without putting excessive strain on the wrists, shoulders, back, or neck.
To hold the flute properly, what you need to go for is a three-point support system. You guide the flute towards you with the joint that joins your left index finger to the hand--that is, the palmar surface of the index finger's MP (metacarpo-phalyngeal) joint (thanks, Larry)--or with your left thumb if you play with a "flat-fingered" or "piping-style" grip. Your right thumb guides the flute away from you and is counterbalanced by your jaw at the other end of the flute. The flute isn't gripped in either hand; rather, it's balanced between these three points. Note that the best place for your right thumb is braced against the side of the flute, not underneath. Many players (myself included) also brace the little finger against the flute. If you do this, you may want to turn your foot joint out so that the E-flat key is out of the way - just be sure that you roll it back in before you start playing a tune that uses that note. You should be able to keep your right wrist fairly straight.
Note that it's best to seal the holes of the flute with the fleshy undersides of your fingertips, not the bony ends, so you can keep your right-hand fingers pretty flat. It's much less of a strain to reach the holes that way, too.
Be very careful with the position of your left hand. The standard grip requires you to bend the wrist backwards a little, but avoid going to extremes. Long periods of wiggling your fingers with your wrists flexed can lead to carpal tunnel syndrome or worse problems. You can minimize the bend in your wrist by bracing the flute against the MP joint rather than the finger itself, and by turning your head a bit to the left. Play around with these variables until you find a comfortable, relatively tension-free position, and check yourself often in a mirror.
You'll also find that turning your head to the left also lets you raise your right hand to playing height without having to lift your shoulder much at all. Both your elbows should be held loosely and comfortably, more or less at your sides.
The flat-fingered left-hand position, with the thumb on the far side of the flute, is a relatively low-tension grip, and much easier on the wrist. However, if you use this technique, you may find yourself getting into a posture that puts excessive strain on your neck. If you have a flute with a two-piece middle section, you can roll your left-hand section outward, which lets the hand reach the holes comfortably without you having to bend your head down to get at the embouchure hole. The only other drawback to holding the flute this way is that you can't reach the B-flat key, but that's not a major problem in the context of Irish music.
Above all, keep in mind that if you experience pain or tension while you're playing, you're doing something wrong. Playing the flute isn't supposed to hurt--that's what dancing is for!