Many German words contain vowel letters crowned by a dieresis, viz. <ü ö ä>, as in führe ‘(I) lead’, Föhre ‘pine’ and Fähre ‘ferry’. In English jargon, the word umlaut is variously applied to the dieresis, to letters provided by this diacritic or to other German letters lacking from an English keyboard. Such usage is non-scientific; you may wish to consult a page on the umlaut, activating your preferred translator if necessary. What matters presently is the pronunciation of these complex letters, as it constitutes one of the thorniest problems for someone learning German as a foreign language.

First of all, the letter <ä> can be done away with as it is simply pronounced [ɛ], a sound present in the native language of most people studying German. What remains are <ü> and <ö>. These are pronounced [y] and [ø]. To be precise, they are pronounced thus if they are long, but [ʏ] and [œ] if they are short. In any case, they are rounded front vowels. The default shape of a front vowel in all languages of the world is with spread lips, as is the case with [ɛ]. The problem is with rounding the lips with a front vowel. Here are the major options of what you can do about it:

Learning the correct pronunciation

Although rounded front vowels are marked against unrounded ones, they are not actually exotic. They occur, among many others, in French, Danish, Hungarian and Turkish. Their articulation does not require special gymnastics but rather the combination of articulatory movements which all of us are familiar with: [y] is like [i], but just only rounded; and it is like [u], but just only front. There are, thus, two ways of approaching the articulation of a front rounded vowel. Both of them presuppose that your variety of English does at least provide back rounded vowels in the example words; otherwise you would have to base yourself on example words of other languages you may be familiar with. Then do the following in front of a mirror:

• To arrive at [y], do either of two things:
  1. Do a [u], as in Engl. you [ju:]. Observe that your lips are rounded and your tongue is back in your mouth. (If you draw it further back, you suffocate.) Now slide your tongue to the front of your mouth (as for [i]) while keeping everything else as before. That is, keep your lips rounded and keep your tongue raised to roof of the mouth. (If you raise it even farther, you start pronouncing a consonant rather than a vowel.) If you do this, you arrive at [y].
  2. Do an [i], as in Engl. meet [mi:t]. Observe that your lips are spread and your tongue is front in your mouth. (If you would raise it farther, you would be pronouncing a [j], as at the start of you.) Now round your lips (as for [u]) while keeping everything else as before. That is, keep the tongue where it is – do not retract or lower it! If you stick to this, you arrive at [y].
• To arrive at [ø], the two options are analogous to the preceding ones:
  1. Do an [o], as in Engl. yoke [jo^wk]. Slide the tongue to the front of your mouth (as for [e]) while keeping everything else as before. You will arrive at [ø].
  2. Do an [e], as in Engl. Yale [je^jL]. Now round your lips (as for [o]) while keeping everything else as before. You will arrive at [ø].

For both sounds, approach #a is merely didactic. As a matter of fact, these sounds are not actually rounded vowels modified by fronting. It is approach #b that corresponds to the essence of these sounds: they are essentially front vowels modified by lip rounding.

How not to pronounce – and to not pronounce – front rounded vowels

You take the second avenue if you are a member of the large group who despairs of the foregoing exercise. The way out chosen by 100% of this group is to regard the dieresis as an extravagance of German orthography which causes Germans to apply some uncategorizable modification to their sounds but is otherwise best ignored. That is, they pronounce <ü> as [u] and <ö> as [o]. This is precisely what you are not supposed to do, for more than one reason:

• The umlaut carries what linguists call a relatively high functional load in the language; i.o.w., there are many occasions where it makes an essential difference. If you ignore it, you create tons of homonyms which will make it harder to understand you. To mention just one important subcase: Many nouns form their plural by umlaut. Thus Mutter is ‘mother’, but Mütter is ‘mothers’.
• By contrast with other phonetic variants – say, pronouncing velar [L] instead of palatal [l] –, you will be using a mode of speaking that no German speaker would ever use. Pairs such as [u] and [y], or [o] and [ø], are not phonetically similar to the German ear.
• You will be branding yourself as somebody incapable of looking beyond the boundaries of their native language.

Here is the correct way of sidestepping the problem altogether: Remember that [y] is similar to [i], and [ø] is similar to [e]. Therefore, omit the rounding of the front vowels, but keep them front.

• Where it reads <ü>, pronounce [i]. Thus, instead of [fy:ʁə] (führe, see above), say [fi:ʁə].
• Likewise, where it reads <ö>, pronounce [e]. Thus, instead of [fø:ʁə] (Föhre), say [fe:ʁə].

This may look wrong at first. There are, however, compelling arguments for it:

• In all of German literature, including Goethe and everybody, the front rounded vowels rhyme with the front unrounded vowels and never with the back (rounded) vowels. In the poem “Freude schöner Götterfunken” (Ode to Joy), Schiller (quoted by Beethoven) writes:
  Deine Zauber binden wieder,
  was die Mode streng geteilt.
  Alle Menschen werden Brüder,
  wo Dein sanfter Flügel weilt.
  
  This just corresponds to rules of German (folk and literary) poetry.
• The front rounded vowels are a property of the German standard. Most dialects have unrounded them long ago (as has done English even longer ago). Therefore, if you say [zi:s] for süß ‘sweet’ and [ʃe:n] for schön, you will at least be speaking (a variety of) German, whereas if instead you say [zu:s] and [ʃo:n], would will be speaking gibberish.

Sum total:

• the workaround for [y] is [i], not [u]
• the workaround for [ø] is [e], not [o].