is there seriously not a plugin or something you can add to Windows to use something logical to type umlauts instead of having to memorize a new keyboard layout, or four digit alt codes? Linux and Mac both have easy solutions to this problem...

There must be a way to do this within duolingo???

If you're on W7 right click the keyboard on your bottom right (taskbar), settings, Add (on the right), search for German (Germany), open the keyboard tab and check German. Click ok. Now if you see the bottom right you should have DE instead of US, that means you have a German keyboard. The umlaut is there somewhere in your keyboard just search for it. To back to US keyboard click the DE and switch to US. Done. To go back to DE, same last step.

I have umlauts on my keyboard, but not assoiciated with any letter. I push that and then the letter i want umlauts over: ie umlaut + a gives ä. Works in windows, works in linux

If you're on Windows and don't want to change your keyboard layout to German (which will swap your Y and Z keys as well as move a lot of the punctuation to keys you might not be used to; see http://en.wikipedia.org/wiki/Keyboard_layout#Germany_and_Austria), you can use "Alt codes" to enter characters that aren't on your keyboard. (Hold down Alt and type digits on the numeric keypad. It's only slightly less fiddly than using the mouse method...)   ä Alt+0228   ö Alt+0246   ü Alt+0252   ß Alt+0223   Ä Alt+0196   Ö Alt+0214   Ü Alt+0220

On Macs, first type "alt + U", then type the vowel you need, and you'll get the umlaut!

In Linux, of course (for those looking for the answer), the best solution is to enable the 'compose key'. Exactly how to do that varies between window managers, but once you've made a key the compose key (I use capslock), you can then press "<Compose Key>, <">, <u>" for a 'ü'. It can do a lot of other things, too. See:

http://www.hermit.org/Linux/ComposeKeys.html