I believe that "achievements" in their most basic form are good for a game. I see achievements as more of unlockables or the feeling of successfully completing a specific scenario.
Take for example Super Smash Brothers Brawl, few people ever mention the alarming amount of work that has gone into the game in terms of music, customization, or even the wall of "achievements" that can be unlocked, enhancing the gaming experience.
For those who would rather label me as a fanboy for the previous statement, I offer the following counter-argument. Take for example the Half-Life series, you start with nothing, then a crobar, and work your way up to bigger weapons and tools. You "achieve" a greater ability to survive in a hostile environment. The same can be said about a lot of FPS games.
Final argument, and although I absolutely despise the game, players of World of Warcraft play the game to "achieve" the maximum level...
As for the simple text and points achievements, some I can agree with (those that are like a badge, medal, or award), but the ones that simply "mark" your progress can go away...