Mountain Lion - Umlauts with english keyboard layout
There are a few means of typing diacritical marks.

Input Sources Method

You can enable a German Language layout by opening Language &amp; Text in System Preferences, going to the Input Sources, and checking the box next to German. This isn't quite the same as your previous set up, but it allows you to quickly type ü, ö and ä with the [, ; and ' keys respectively. If you check the Show Input menu in menu bar box, you can quickly switch between different layouts, if you want the standard English layout most of the time. The menu also lets you open the Keyboard Viewer, which will show you all the changes of the German layout.

Create a Custom Keyboard Layout

If the German Layout isn't exactly to your taste, you can use Ukelele to create a custom layout.

Open Ukelele, and click File → New From Current Input Source to start a new layout based on your current one. To make the ' key be your umlaut key (so that you press it, followed by a vowel to create the character), click the Create button, click the ' key (or whichever key you want; you can add modifier keys if you wish), then in the dialog that pops up, select Choose an existing dead key state and select State 4. If you want to rebind another key so that you can still type the ' character, double click the key you want to change, and change the output to '.

You'll then want to change the name of your layout so you can distinguish it. Set Keyboard Name… under the Keyboard menu will do this for you. You can also set a custom icon if you want.

Once you've finished with your layout, save it to your the ~/Library/Keyboard Layouts/ folder (or the /Library/Keyboard Layouts/ folder if you want it to apply for every user on your system). You should then be able to select it in the Input Sources section, as above.

Other Methods

By default on Lion and Mountain Lion, you can hold down a vowel to get a pop-up menu that gives you the various diacritics you can apply to that letter, like so:


You can either click on the one you want, or type the corresponding letter. This functionality is enabled by default, but can be disabled by running defaults write -g ApplePressAndHoldEnabled -bool false in Terminal. To enable it, replace false with true.

The other common method, as you have discovered, is to type option ⌥u, followed by the vowel you want. This works on OS X going back to at least 10.4, possibly earlier.