There are some minced oaths in German as well, such as Scheibenkleister or Scheibenhonig. They might be less offensive than Scheiße (or Mist), and, at least in the case of "Scheibenhonig", might even help brighten up the general mood a bit.
Apart from that, Verflixt (und zugenäht [optional])! would be another not-quite offensive alternative I can think of right now, this is probably some less variation of Verdammt! (en: damn!).
Edit: Herrgott nochmal! (probably more offensive among very religious people) and Das darf doch nicht wahr sein! are no proper swearwords, but might help releasing some steam.
In Northern Germany, using Schiet (Low German for "Scheiße") in various forms is not considered very offensive either.