I can't speak as to how to individual particular programs work, but at least as of a year ago, most of the good AIs used something called Monte Carlo Upper Confidence Trees.

They're, unfortunately, not based on intuition. Just statistics.

Basically, the machine plays many, many random games. The more winning games which play a particular stone, the more valuable that particular position is. Then the position with the highest value is chosen.

This alleviates the need of brute force search (which is just too large for Go).

As far as I understand, most of the more successful AIs use UCT for the general game, but then fall back to heuristics and brute search for small, local conflicts (like if forced some stones to be played until death) and counting stones.

Sensei's library has some nice high level information written about the topic: http://senseis.xmp.net/?UCT

More recently, DeepMind (and someone else independently at Edinburgh, I believe) has published a couple papers about training a neural network to play by teaching it to predict the next move of pros given the board state. These use modern computer vision techniques and data. Last paper I saw, this did much better than GNU Go (which doesn't use UCT, and is very weak), but still not as good as monte carlo based methods. There has yet to be a combination of the Neural Network and Monte Carlo methods, but they're quite complimentary.

I believe we'll see that combination in the next year or so, and this will nearly close the AI-human gap.

Edit: Monte Carlo. :P Hooray for autocorrect.