German quotation marks are broken in XeLaTeX

Never use babel with XeLaTeX. For internationalization support use polyglossia instead. Besides, in this case it's not even needed. You can enter the quotation marks either using ASCII symbols or direct UTF-8:

\documentclass{article}

\usepackage{fontspec}

\setmainfont[Ligatures = TeX]{TeX Gyre Pagella}

\begin{document}

<< >> ,, ``

« » „ “

\end{document}

polyglossia also has some babel shorthand support to ease the transition from babel.

Update (March 2016)
babel now supports XeTeX so the following will work:

% !TEX program = xelatex
% !Mode:: "TeX:UTF-8"
\documentclass[12pt,ngerman]{article}

\usepackage{babel}

\usepackage{fontspec}
\defaultfontfeatures{Mapping=tex-text}

\begin{document}
These are German \glqq quotation marks\grqq.
These are German "`quotation marks"'.
\end{document}

---

Original Answer
Think you’d better use polyglossia instead of babel:

% !TEX program = xelatex
% !Mode:: "TeX:UTF-8"
\documentclass[12pt]{article}
\usepackage{polyglossia}
    \setmainlanguage[spelling=new,babelshorthands=true]{german}
\usepackage{fontspec}
\defaultfontfeatures{Mapping=tex-text}
\setromanfont{Times New Roman}
\newcommand\glqq{"`}
\newcommand\grqq{"'}
\begin{document}
Thess are German \glqq quotation marks\grqq.
Thess are German "`quotation marks"'.
\end{document}

(I changed the font …)

When using babel you should check if the language definition files you want to use are already adapted to the unicode engines -- this isn't the case for a number of non-western scripts -- but with (n)german it will work fine with xelatex. You are not getting the quote because xunicode doesn't provide the necessary definitions. It is easy to add them. E.g.

\documentclass[12pt]{article}
\usepackage[ngerman]{babel}
\usepackage{fontspec}
\defaultfontfeatures{Mapping=tex-text}
\setmainfont{TeX Gyre Pagella}

\DeclareUTFcharacter[\UTFencname]{x201C}{\grqq}
\DeclareUTFcharacter[\UTFencname]{x201E}{\glqq}

\begin{document}
These are German \glqq quotation marks\grqq.
\end{document}

Also take a look at the package csquotes

(Edited on march 2016 to adapt to the new commands in fontspec and the development in babel.)