You may leave out the funny dots if not available: Jan Jurjens
In German it is customary to substitute ü by "ue" when necessary: Jan Juerjens

In .html my name is written Jürjens, in TeX J\"urjens.
Unfortunately, bibtex doesn't properly handle non-ascii letters when creating citing abbreviations in the alpha style, so there you need the slightly complicated-looking:
author = {Jan {J}{\"{u}}{r}jens} (except if there are at least three authors to a publication; I promise I try to make this happen more often to simplify things).

Note for searching citations etc. in public databases: Citeseer does not seem to be able to deal with the Umlaut in my name. Google scholar does a slightly better job at this but only using this search fixing an apparent OCR problem (and then one still needs to add up the citations for the different spellings for each document...).

Popular misspelling: Jan Jürgens (with variations Jan Jurgens and Jan Juergens).

By the way, in German, Jan is a male first name (as opposed to the English female usage).