First off, the situation you described makes those runners seem like very poor base-runners. A runner should never tag up from 3rd on a line-drive back to the pitcher (let alone any line-drive hit within the infield) ESPECIALLY with 0 outs. The correct action to take is for the baserunners to freeze until they see the whole play.. if the ball is caught go back to the bag, if the ball is dropped then either run or stay.
Actually, I am editing my initial answer... I didn't read the whole scenario (the last sentence).. if the runner who touched home beat out the pitcher turning a triple play.. then this rule is not applied and the run should count.
A run is not scored if the runner advances to home base during a play in which the third out is made (1) by the batter-runner before he touches first base; (2) by any runner being forced out; or (3) by a preceding runner who is declared out because he failed to touch one of the bases.
(1) the batter-runner never got a chance to touch first and was indeed out before the runner who was on 3rd touched home, but the batter-runner wasn't the 3rd out..
(2) none of those runner's were forced to go the next base once the ball was caught by the pitcher
(3) I am assuming that all runners touched all of the bases (well at least the bases that they got to, and the preceding ones)
So I would say that run did count.