Probably the simplest way to achieve this is with git archive. If you really need just the expanded tree you can do something like this.
git archive master | tar -x -C /somewhere/else
Most of the time that I need to 'export' something from git, I want a compressed archive in any case so I do something like this.
git archive master | bzip2 >source-tree.tar.bz2
git archive --format zip --output /full/path master
git help archive for more details, it's quite flexible.