Although I insist on my vote of Flash (Jay), Green Lantern (Alan) and Hawkman (Carter) for the reasons I gave before, when I started reading the Justice Society (around the time of All-Star Comics and their short stint over at Adventure Comics), Dr. Fate (Kent Nelson) had a much larger role to play on the team than in previous and later runs. He was like the mentor, elder statesman, of the team (Carter still was the leader, though), guiding them on their path.
And the Trinity of Flash, Lantern and Hawk was still very evident in James Robinson's version of the Society in the New 52's Earth Two, only it was Flash (Jay), Green Lantern (Alan) and HAWKGIRL (Kendra) , for there was no Hawkman in Earth 2. These three were the central figures of the whole series.
As for a modern Trinity for the Society, it's harder, for there are many important members to the team ever since the JSA's relaunch started them going full strength again (Sand, Power Girl, Wildcat, etc...). But I think that the three that best embody the future of the JSA are/were - Mr. Terrific (Michael Holt), Starrgirl (Courtney Whitmore) and Dr. Mid-Nite (Dr. Pieter Cross).

Peace