Classical players usually don't "get" it -- "it" being the lilt and lift and ornamentation that makes Irish tunes, in particular, so wonderful. The same is true for violinists playing fiddle music (generally speaking; I'm sure there are exceptions).

On guitar, Tony McManus is definitely the 500-pound gorilla these days. Check out also Martin Simpson, Chris Newman, Americans El McMeen and William Coulter.
That's if you're talking about Scots and Irish dance tunes played solo on guitar -- which doesn't really do it for me, brilliant as these players are.

I prefer the tunes played on fiddle, flute, box, pipes etc, with guitar accompaniment. Malinky's Steve Byrne is great in that department; so is Donogh Hennessy, former Lunasa guitarist; Americans Matt Heaton and Ray Frank (Mutcatter DADGBE); John Doyle, Ged Foley... the list goes on.

Cheers,
Michael