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/to/zipfile.zip master
git help archive for more details, it's quite flexible.
Be aware that even though the archive will not contain the .git directory, it will, however, contain other hidden git-specific files like .gitignore, .gitattributes, etc. If you don't want them in the archive, make sure you use the export-ignore attribute in a .gitattributes file and commit this before doing your archive. Read more...
Note: If you are interested in exporting the index, the command is
git checkout-index -a -f --prefix=/destination/path/
(See Greg's answer for more details)