It's a pain when you haven't got a German keyboard to figure these characters out.
Is it acceptable from a style perspective?
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You can do it in every-day conversations by E-Mail or chat. But when writing a somewhat official document, you really should try to get the umlauts right. It's just a question of conformity: You want to use the language, so use it correctly.
A whole different problem that will probably come up if you manage to get the umlauts in your email is encoding - if you write your E-Mail it can happen that your encoding doesn't know these letters so it will print some black squares or question marks. To avoid this, you have to check that you use an encoding format that "knows" umlauts, like UTF8 or ISO-IEC 8859-1.
If you can't manage this and have to write an extra "e", make sure you add a line about it saying you're sorry for the bad formatting and that it's because you have trouble getting the letters to work. Just for safety so nobody can blame you for it afterwards.
Try to omit this practice as much as possible.
Nur sehr geubte Leser werden diesen Satz so entziffern konnen wie er ursprunglich gedacht war.
Nur sehr geuebte Leser werden diesen Satz so entziffern koennen wie er urspruenglich gedacht war.
Nur sehr geübte Leser werden diesen Satz so entziffern können wie er ursprünglich gedacht war.
Although this sentence has only three umlauts the third variant is the best to read. Most natives on a "wrong" keyboard use the second possibility, this means that people are used to it. Although its really awkward to read it in my oppinion.
The first possibility simply looks like you don't even know the very basics of transliterating. This might not be of a problem if you're talking to <20yrs old who aren't used to transliterating anymore (Thanks UTF8 :) ), but people aged older will be disappointed.
If you're writing German on a regular bases you should look into changing your keyboard layout to German (link for win, you can do that with every keyboard on every operating system, just google it).
On German Language & Usage-Site
I highly suggest to use new orthography always, there is a vote going on on meta about the general usage. It tends to favour the new orthography by 10 votes difference on 11 votes cast in total.
Back in the days of typewriters this transcription was very common, it's certainly not wrong but the excuse that your computer keyboard does not have those keys is hardly valid nowadays as you can easily change the layout. So if you write a very important document and you use the transcription the recipient may take that as laziness on your part.
I for one use the transcription the vast majority of the time but i do not dare to use it in any paper, i have a custom layout for those where i can write the Umlauts with the modifiers Ctrl + Alt.
Ctrl + Alt + A is equivalent to Ä,
Ctrl + Shift + Alt + A is equivalent to Shift + Ä for capitals etc.)
Only if your keyboard doesn't let you enter umlauts. (Even with a US keyboard, you can enter umlauts. Hold down the left alt key and type "129" on your numeric keypad to get the u-umlaut. Learn the numbers for them all, put them on a little post-it note, and stick it to your keyboard. there's only 4 letters in the German alphabet that don't exist in the English alphabet, so you can learn them pretty easy.)
It's a lot easier on US Mac keyboards to get the umlauts—just use the Option key + u to get an umlaut, then type the vowel you want under it: [Option + u, o] gives ö, and [Option + s] gives the "ß."
However, I have seen a number of German friends and colleagues write emails to me using the "ue" instead of "ü." So while the new style is inarguably preferable, it's by no means the only one in widespread use.
Edited to add: With the release of Mavericks, a new way of typing such characters is now available on English keyboards. Simply holding down a letter for a second or two brings up a menu of options. You can then choose the number corresponding to the desired accent. For instance, to type ü, you can hold down 'u' and then press '2'.
I swear by a fantastic little app called DE Key, you can get it for free from this website: http://german.about.com/library/blcomp_dekey1.htm
It is a safe app developed by an English speaking programmer who started learning German and, like you got frustrated at not being able to type the German umlauts. It is very simple to use as all that one needs to do to access the umlaut, is to use the ALT key before the letter in question e.g. for ALT + a = ä.
As far as I know there is a Mac version too.
The web address provided is not the original website, a longer search would get you to that if you prefer.
Hope this helps!
Maybe I'm not getting it, but surely we don't need to remember the numeric codes? Doesn't the pictorial method work on all keyboards? the umlaut is like a colon: sideways, so you type Ctrl_SHift, then poke : , then release ctrl-shift, then type a: aaa? nope, it doesn't work here. But if I type it in Microsoft Word, it works, and I can paste it here: They’re all pictorial; ^ gives the hât, ~ gives the tilde õ , ‘ (just Ctrl – ‘ then e, no shift) gives é, etc. You can pretty much guess them….I can’t remember any more right now. How about the backwards acutè? Yes, it’s just below the tilde on my keyboard, again without the shift?