What I did, when I had to program AI for some simple game, is have the AI player build a game tree. Depending on how simple the game is.. there are different techniques. I represented my game tree with a 3 dimensional array of around 30 elements. The game was simple enough where I could implement the entire tree in memory, but for advanced games, like Chess this is impossible.
For a game like tripple triad you'd have to calculate different states and put these states into a game tree like structure. The computer would calculate these states and mark them based on who is likely to win. E.g. "If I play this card here.. can I possibly win?? No? Then what if I put it in the next sqaure?". It would then continue to expand the nodes and build the tree further in areas where it had a better chance of winning since the AI player wants to go down that branch of the tree.
I know this probably doesn't make any sense, but if you can get an AI book you'll want to look into some of the MIN-MAX algorithms and ALPHA-BETA tree pruning. The books I have use the classical tic-tac-toe example to explain how they work but you should be able to adapt it to tripple triad.
At the beginning of the game there will be a lot of stuff to calculate so you could make it a little faster by forcing the computer to play a cards that should be played in certain spots..
e.g. if you represent a cards numbers by {Left,Top,Right,Bottom} and the AI has a card like {0,0,A,A}, you would force him to play that card in the top left corner.
Hope that helps.