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

ZIP archive:

git archive --format zip --output /full/path/to/ 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)