What causes the popup alternate characters from showing?
Are there any background applications or kernel extensions that could disable the popovers? For example KeyRemap4MacBook disables them.

What input method do you use? The popovers don't work with all input methods, or custom keyboard layouts that are not saved in the bundle format.

You could also try to run defaults write -g ApplePressAndHoldEnabled -bool true and reopen applications. Or delete ~/Library/Preferences/ByHost/com.apple.HIToolbox.*.plist and restart.

Does it work on other accounts, or did it work in earlier versions of OS X? Is there a process named PressAndHold?

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Character Picker stopped working in OS X Lion