Why the greenish tint in the Matrix trilogy?

The green tint is suggestive of the early monochrome cathode monitors, a piece of imagery associated with early computing:

The Wachowski's wanted to create a mood palette, so they could cross cut between the Matrix and the Real world and an audience could immediately identify where the action was occuring through the use of such a colour scheme.

This was discussed by the notoriously secretive Wachowskis in one of the very few interviews they did (at the insistence of their distributor) in order to promote The Matrix.

WachowskiBros: One of the things we tried to do with the Neb for when we were shooting “in the real world” was use long lenses to separate the humans from the backgrounds as opposed to when we shot the Matrix we used short lenses to place the humans in specific deep spaces. We also tinted all of the lights blue in the “real world” and green in the Matrix.

...

Spark: What made you decide on the green tint for being in the Matrix?

WachowskiBros: It was a whole motif inspired by the phosphorous green of old PC’s.

I like the answer @JohnSmithOptional gave. Also, the directors have explained that they thought the color blue was too happy/cheerful, so they chose to make the sky within the Matrix green. The one time that bright blue is used is in the last half hour of the 3rd movie: a ship flies way up above the clouds, in the Real World, and there's a moment of beautiful blue sky. I think Trinity or someone gasps. Then they fall back through the clouds into the darkness to battle the Machines.

(I'm sorry I can't cite the source where the directors said that, but I'm SURE I've heard or read it. Might have been on the DVD bonus material.)

I have an additional reason to add (though I agree that the primary one is due to the greenish tint of computer screens like VT100s). The physical city where they shot The Matrix, Sydney, is also known as the Land of Oz (Oz, short for Aussies, short for Australia) and much of the city uses green marble as a tribute to the Emerald City of Oz.

This made much of the shooting of the scenes within the Matrix already inherit a greenish hue (think of the twins and the gov't lobby scenes). I suspect the Wachowskis might have been subtly conscious of this. Green probably just felt right to them for many reasons and it certainly plays perfectly into the movie as a whole.

There is far more symbolism in the Matrix than most people realize. The green tint is symbolic of the basis of a system of control. It permeates our world today. Those who control its issuance have total control over just about everything - including our governments and political process. It is the lie - the nightmare - that the vast majority are unaware they are a willing participant in.