After 7 months of typing “rueschlikon” to find and copy-paste “Rüschlikon” when filling out address fields (that’s the name of the community in the canton of Zürich where I live now), I finally slowed down to think if there was a better way. Of course, there is: Option + u + (Shift if needed) + letter itself. And voilà: ä ë ö ü Ä Ë Ö Ü.
And yes, you guessed it, I used Option + ` + a to type à.
Jenia Laszlo 
The other alternative would be a long press on the “u” key, until you get a list of alternative “u”. It took more than a year to discover this option 🙂
Húúú wow! That is pretty handy, too, thanks for the tip!
This is super handy and was very necessary for me when I switched to Dvorak but still had to type Swedish. There are also lots of characters that can be typed by using only Option + letter, for instance ‘å’ or ‘ø’, but maybe you already know about this.
Turns out that å and ø are the characters I don’t use, because they are not in the German, French, or Hungarian alphabets (that I might be typing). Good to know for troubleshooting i18n issues, though!