If your last name were Dume, pronounced “doom”, could you place the two dots above the “ü” [Düme] so people would understand its pronunciation. Will that be riduculous usage? Just leave it be?
[last name variation]
If there were two dots over the ü, I’d be thinking that it was a German word, with German pronunciation rules, so the ü would be pronounced with a vowel sound that is not found in English but is like the “u” in the French “une”, and the word would have two syllables.
Why would you need to indicate pronunciation? If you’re writing a script or something and you need to show pronunciation, I’d just do it like this:
Dume (DOOM).
Robin
Wouldn’t most English-speakers assume that “Dume” is pronounced “DOOM”? It seems like it would mainly be a problem if you wanted it to be something like “Doo-may.”
If anything, you’d indicate a long “u” with a macron (horizontal line), not an umlaut.
Essentially only North Americans would pronounce it that way. The rest of the English speaking world could easily assume it was equivalent to “dune” or “consume”. Those words are not phonetically “doon” or “consoom” but rather “DYOON” and “CONSYOOM”. So for a sizable chunk of English speakers “Dume” would be pronounced “DYOOM” by default.
As Giles pointed out the “ü” normally indicates a completely different phoneme from a “u” (sans dots). “ü” is pronounced with your tongue in the position for an /i/ (that is to say the “ee” in “peek” or the “ea” in “mean”) but with the lips rounded as for an “o” or “u”. To my ear this sounds kind like cross between the two sounds but not exactly.
In any case I would not reccommend that you attempt to spell it this way. I don’t see any good way to get around the problem of trying to indicate a pronunciation of “Dume” as “Doom” unambiguously without simply spelling it “Doom” or maybe “Dum” with a macron (but I’m not sure how many people will get this). There really isn’t any reliable way that I know of to indicate that the final “e” is voiceless (short of just dropping it).
My reccomendation, if you are committed to an “e” on the end, would be to just spell it “Dume” and be ready for some people to mispronounce it (or be uncertain of pronunciation). I think many people will see the analogy to “dune” and pronounce it like that.
LATE BREAKING NEWS FLASH!
Actually a second course suddenly occurs to me. I think this might work better, though I may be wrong.
You could spell it “Doome”. I think most people will see the “Doom” in there and perhaps have a better chance of pronouncing it as you intended. The advantage here is that a double “o” is pretty firmly established as a “u” sound in English, compared to a single “u”, which has a myriad of pronunciations, most of them not the classic “u” of most other languages.
I’d pronounce it ‘dummy’.
Just 'cause that would be more fun.
How are you today, Mrs. Dummy? And all the little Dummy children?
A) “Dume” I’d pronounce “Doo-MAY”, but probably ask for clarification first.
2) “Impeach Bush” car decals are the google ads for this thread… Sorry, I just find that funny.
If my last name was Doom, I’d expect to have a Castle/Lair, rulership of a small European country, cool metal armour, nemesii in the form the Fantastic Four, and the prefix “Doctor”… 
Seriously though, if I saw the last name “Doom” in print form with no other reference (say, on a resume or as a mailing address), I’d assume it as pronounced “Doom”, like the fate (or the computer game, depending on your age!).
Similarly, I’d expect Dume to be “Dumay”… you get the idea.
Captain’s Log Additional: I’d pronounce “Dumay” as if it were French, not as “Dummy” (or something similar). :smack:
Confound the lack of an edit function! 
Oh, I see. Just like “dune” is pronounced “dunay.” If I encounted someone with that name I’d ask them how they pronounce it. If I saw it written it would be, mentally, Doon until I learned otherwise.
Wouldn’t it be spelled Dumé?
Shaping my mouth like the vowel (ä, ö, ü – I don’t remember any words at the moment with ë) and saying ‘ee’ is how I learned it in my German classes.
I’d go with Doume, or maybe Duume. I assume that you don’t want to change the actual letters, as it confuses governments and computers, but:
Dúme looks like a Spanish word (DOO-meh)
Dûme looks like a French word (DÜM, one syllable with the rounded Fr. / Ger. [ü])
Düme looks like a German word (DÜ-meh)
Dùme doesn’t look like anything, but doesn’t help either, and all the others are too hard to type.
Důme? Confusing enough that it might just make people ask every single time…