In order to get it looking just right, I've been doing a little research to see what techniques were really used on these garments in period. Somewhat to my dismay, I'm finding out that some rumors I've heard about the modern leine's authenticity are true.
First of all, it's interesting to know that sumptuary laws around the time of Henry VIII required that the Irish not wear any more than roughly 8 yards of fabric in their shirts. Apparently, this was a great hardship, and was considered evidence of the Englishman's desire to keep the Celt down, man. But, seriously! The one I'm making is huge-sleeved, and pleated and gathered all over the place, yet contains only about 5 yards. I'd consider this evidence that period leinte tended to be very, very full and voluminous.
Here's the thing, though: the distinctive, pleated sleeve reenactors have come to associate with the leine was apparently never gathered or pleated in period. Every source I've read says that the gathers (and more recently, the draw-string, like I'm using) originates from Ren Faires in the 1960s and 70s. It was just an expedient little anachronism to keep your sleeves out of the way on hot days at Faire.
But check out this drawing from the Tudor period, of Irish infantry soldiers. Look at the guy on the left. Doesn't that sleeve look pleated to you? If it's neither gathered nor pleated, I'd like to know how those guys kept their sleeves from seriously inhibiting their sword-arms.
So far, I can find no explanation for how these enormous garments kept from sliding down all over the place. I will keep searching for information, and I'll post again when I know. I'm sure you all will be on the edge of your seats until I find the answer!