I'll go with the opening, since the opening sets up both the middlegame and the endgame.
Most Important Part Of The Game

I agree borgqueen, I thought the statement was the other way round, like...
Good endgame come from the good middlegame and the good middlegame come from the good opening.

Middlegame.
It's not the opening, because few games between equal (non-GM) opponents are decided in the opening.
It's not the endgame. Although the endgame decides the game, and mistakes there happen often and swing the probable result of the game around all the time, too few games between amateurs reach an endgame to be the most important phase.
So I say the middlegame. Most games are decided there.

If "important" refers to what part to study, then the endgame.
The endgame teaches you the power of the pieces, which can be applied to all phases of the game. Plus, the rules of the endgame never change. Opposition will never and can never be substituted for a "better" method.
Middlegame comes next, then the opening.

In terms of winning the game, they are all equally important... if you are poor in any area, it will still result in the same score- one point against yourself.

The most important part of the game is checkmate.
By the way, the soviets considered the endgame the most important and taught chess from the end backwards in the soviet chess schools.
I think that the endgame is where most players make mistakes because they don't get that much practice with it. The endgame would seem to be the aspect of the game that most players need to improve upon through study and practice.

No offense intended NOLAUPT, but questions like this imply the need to place one aspect above all others (perhaps your intent is to get answers that aren't so simple).
In life and on the chessboard, this is a mind-crippling fallacy. Those who can see every aspect for its inherent flaws and benefits (depending on the situation) will usually triumph over those who think in more black-and-white terms. For example, it's the same problem with those who say the bishop is better than the knight and vice/versa. Analyzing a position will show truth to either statement, but only on a case-by-case basis.
With all that said, I'll agree that no phase of the game is the most important.

NOLAUPT
Wow I can't believe only Scarblac suggested the middle game.
I think, ultimately, the middle game is the most important phase, because that's where the vast, vast, vast majority of all games are won or lost.
Openings are very important, and become much more important when you become a stronger player, but I'm very confident that the average player can reach an 1800-1900 rating here without any studying of openings at all! Why? Simply put, there is no opening in chess that forces Black to lose material! If you think about it, if there was a "Black Always Loses a Pawn In This Opening" Opening, then everyone everywhere would play it all the time as White in tournament play. However, to this date, no such opening exists. The best thing you can hope for in the opening, when you play someone who is approximately your skill level, is that you achieve a +/= advantage as White or an = or =/+ advantage as Black.
A beginning or intermediate player will achieve the said +/= or even +/- advantage, only to blunder a pawn or lose to a Knight fork in the middle game about 5 turns later.
Think about it, let's say you got to challenge Garry Kasparov to a game of chess where you play White. The game proceeds as follows:
1. e4 Nh6?
2. d4 Ng8??
Now the grandmaster just wasted his first two turns, putting you ahead a full 3 turns in development. Garry has just failed the opening just about as bad as you possibly can, but I'm pretty sure he would win/draw in the following game against pretty much everyone here due to his strong chess ability that allows him to outplay you in the middle game despite a bad opening position.
Next, the endgame is very important. And it's the area in my game which is lacking the most, so I'm here in Starbucks studying Silman's Endgame Course as I write! But over the course of the past several months when I've been noticing an improvement in my game, it didn't really matter that my endgame was atrocious. Really, if I entered the endgame up a piece, I was guaranteed to win/draw 100% of the time anyway!! So it didn't even matter if I knew concepts like "Indirect Opposition", "Outflanking the King", and the "Lucena" and "Philidor" Positions. Simply, the point is that all the endgame theory in the world doesn't matter if you blunder a piece, or in some cases, even just 1 pawn, in the middle game . . . or opening :)
For everyone rated 1800 or lower, 80-99.999% of games will be won or lost in the middle game. If you know how to play a good middle game, you should, by definition, be able to play a very solid opening game. The two phases aren't all that much different.
If you can prevent yourself from falling behind in tempo and guard yourself against dropping pieces/pawns in the middle game, then why shouldn't you be able to do this in the opening, if you really think about it?
But yes, once you become a more intermediate/advanced player, then all phases of the game begin to become extremely important. It's tough to play a good middle game if you exit the opening phase with a position that you are uncertain about and/or uncomfortable with.
And the tactical mistakes and blunders that you find you opponents made on a regular basis when you were facing 1300, 1400, 1500, and 1600s become fewer and far between when you start challenging tougher competition. More often, you'll find yourself reaching "complicated" King and pawn, and Rook and pawn endgames that you may be a little clueless about (like me).
However, for the serious beginner or intermediate player intent on working hard and improving his or her skills, then the middle game is the most important thing to master. Of course, this is easier said than done, as even grandmasters and World Champions still make mistakes in the middle game, proving that chess is one of the most fascinating and dynamic games the world has to offer!
Opening, MiddleGame Or End Game