IT WAS pandemonium. Or as close to that condition as the louche world of jazz can get in the mid-afternoon.

With the sun still high in the Dublin sky, one day back in May 2000 when we were all still young and innocent, actual queues formed outside Vicar Street for a matinée performance by American pianist Brad Mehldau and his trio. Yes, a matinée, hastily added to Mehldau's tour schedule when it became apparent that one concert was not going to be enough to satisfy the demand for tickets.

Jazz is not supposed to be like this. By most definitions, jazz has not been popular music since the late 1940s, and Brad Mehldau, for all his talent and good looks, is certainly no pop star. Who knew that this many people had even heard of him, let alone that they would give up a perfectly respectable spring afternoon to enter the chiaroscuro of an auditorium to listen to him make stuff up on the piano?

Nowadays, Vicar Street has been sufficiently enlarged as to obviate the need for matinées, and jazz has slunk back into the shadows from whence it came. But Mehldau remains the great white hope of the jazz piano tradition and one of its genuine celebrities –articulate, urbane and unusually presentable for a jazz musician. You would bring Brad Mehldau home to meet your mother. And, as that lucky matinée audience discovered, he can really play the piano.

"The matinée is, in general, not my favourite time to play music," says Mehldau, remembering that day and looking forward to his evening concerts in Dublin and Cork next weekend. "The blood sugar is still adjusting, and somehow, one feels more tired earlier in the day than later. Still, once we started playing, it didn't matter. Vicar Street is just a very hip room." Shucks, Brad.

Born in Florida in 1970 but raised in Connecticut, Bradford Mehldau is a typical middle-class New England musician. The son of a doctor, he began playing classical music at the age of five and was regarded as a prodigy in school, but at 13 he discovered Keith Jarrett's Köln Concerts and switched to jazz. By 18, he had relocated to New York to study with pianists Fred Hersch and Kenny Werner, and by his mid-twenties, he had formed his own trio, been signed by a major record label (Warner Brothers) and begun recording a series of albums entitled The Art of the Trio. So far, so breathless. Looking back now, does he think he was too young?

"You know, the twenties are not so young. Young to me is 14 or 15. If you are worth anything, then you should have your own voice by the time you're 25. Look at all the greats – it's always the case."

With self-belief clearly not a problem, Mehldau soon had an international career the envy of musicians twice his age, and the albums – The Art of the Trio, Vols 1-4 – were seized upon as works of genius. Does he wish he had taken things a bit slower and maybe served an apprenticeship with more senior players?

"I wouldn't change a thing," the pianist responds, perhaps a touch defensively, "but that's more just a way of looking at life. I did have a fair amount of apprenticeship, but it wasn't recorded as much. I played with singers and horn players from the time I was 14 and learned how to accompany that way. For me it's an absolute necessity to be a sideman."

Whatever the route, by the beginning of the present decade Mehldau had become one of the most respected and influential jazz musicians of his generation, and his trio, which features bassist Larry Grenadier and drummer Jeff Ballard (who replaced Spaniard Jorge Rossy in 2005), had achieved what for jazz musicians is that most elusive of prizes – a popular following.

In part, that has been achieved through constant touring and the support of a major record label. But it's also because Mehldau seems to embody a connection between the jazz tradition that he is coming out of – Bill Evans is routinely cited – and the young audience to whom he is playing. Incorporating tunes like Radiohead's Paranoid Android and even Oasis' Wonderwall into the trio's set has drawn listeners from outside jazz without ever compromising the sheer daring of the band's improvisations. In a live setting, there are few experiences as intense or as instructive as hearing Mehldau, Grenadier and Ballard deconstruct a familiar tune like Paul Simon's Fifty Ways to Leave Your Lover.

"It is gratifying for us, quite simply, to have the attention focused on us. And so we want to please the people, plain and simple. That does not mean making concessions, but it means that we hope that in making an honest musical statement, we will reach the audience."

Reach they do. Mehldau has been visiting Ireland on a regular basis since 1997, and his relationship with the audience here has grown over the years. He seems to particularly enjoy the Vicar Street crowd, and is relishing his first concerts there since 2006.

"That's a music crowd, plain and simple. Probably with mostly a focus on other music than jazz, but how do I know? I wouldn't want to assume. I just know that the vibe I get there is of people listening with an open mind."

As an improviser, he knows that jazz is all about the dynamic between the musicians and the audience, on which subject he is refreshingly frank: "Sometimes it doesn't work. Usually there is at least a percentage of people in the audience who get what we're doing, but it's hard to generalise. If you've ever been to a concert and then asked more than one person what they thought of it, then you know that you can get wildly different accounts of what just took place, and wildly different responses, from almost ecstasy to torture. I understand, believe me. If you're not feeling jazz, it can really be like, 'Get me out of here now!'"

Not likely. As Brad Mehldau approaches his fortieth birthday, the days of the matinée idol may be behind him, but my guess is that his affection for the Irish audience will be enthusiastically reciprocated. And now would not be a moment too soon to book a ticket.

The Brad Mehldau Trio play Vicar Street, Dublin on Saturday 17 October and the Opera House, Cork on Sunday 18 October. More info from www.note.ie