Well, sometimes. Kinda like sports games, I don't really play conventional shooters. If they have a neat quirk or fun mobility system, I'll be down.
Had a lot of fun with Titanfall because of this. You weren't really restricted by how you had to move and, though the FPS aspect wasn't all too eccentric, the mobility alone reached into other areas and made it interesting. It helped that it worked so well with clasping onto enemy titans- never used my own.
Other recent FPS'...let's see...well, SUPERHOT immediately comes to mind. The FPS that only moves when you do. Granted, I dunno if it's worth the price, but the story's interesting, the presentation's interesting, the concept's interesting, and it's dynamic- something that is essential for a game that has to have limited content. I'd recommend it on sale.
Other than that...looking forward to Doom. Galgun's an On-Rails shooter which, by extension, makes it an FPS, and that's not half-bad. And also, Resident Evil Umbrella/Darkside Chronicles. And, of course, Pokemon Snap. Definitely one of my favorite FPS titles.
Then there's Tribes: Ascend, which was fun for a time thanks to it's sledding mechanic. Vanquish, I hear, is a pretty unique and solid title, so I'd love to check that out. I've been a Halo fan since I was a young'un, but...well, I don't really have the time to write my thoughts about it now. Finally, easily my favorite FPS, Goldeneye 64. It's not modern, no, but it does a lot of things that still make it anomalous for all the wrong reasons. I love the hell out of that game for reasons wholly separate from nostalgia. It's control scheme was...difficult (but considering the controller, I don't blame it), there was a lot of cheese (though, to its credit, the direction could be pretty damn solid for what it was), and it was the reason we have the fantastic Get Down meme (a definite pro), but it had a solid score and a fun, open story mode that is specifically the thing that I love about it. I do hear people reminisce about the multi, which was fun, but for me, my love for the game is all owed to the storymode.