If you're struggling to get this achievement in Career, then there's a very easy way to do this in Grand Prix mode.
Pick any track/car you like, but when you get around to setting the rules, make sure damage is set to at least full (you might get away with partial, but I haven't tried this). Make sure the Rules & Flags are turned off too. Also, go to your Assists and set Shifting to Manual. Everything else is up to you, but for those who want to get it done quick, set the race length to 5 laps.
Once the race starts, begin shifting up and down like crazy using the A and X buttons. This will wear the gearbox very quickly. You can check the wear on the gearbox by pressing B until you get to the last tab.
Just race along the track, shifting up and down as you go. You'll be aiming to get to at least 70% by the time you get to the last lap. If you get to the finish line and your gearbox isn't missing a gear, park up just before the line and continue shifting up and down until your engineer tells you that one of the gears is missing. Then, simply cross the line, and achievement unlocked. :)