
For me, it depends on the game. Some games, like the Ratchet & Clank games, make going after the achievements fun (for the most part). Some games, like a lot of rpgs, the achievements seem like they're tacked on just because the game needed to have them. I've learned to go for the ones that are either fun or increase enjoyment of the game and ignore the rest.
I think most of us have experienced "Achievement Fever" where we obsessed too much over achievements and lost sight of the game at least once.