The Dev team just introduced a new feature within the last patch or two called Community Finder. You can use this feature on the Lodestone to not only join a Free Company of your liking but to also search for Linkshells that may interest you.

Otherwise, there is a shout feature in game. If you AFK for a bit in high traffic zones like Limsa Lominsa, you're bound to find that there is a FC or two that are recruiting.