As of Git 2.15, git diff now supports detection of moved lines with the --color-moved option. It even detects moves between files.
It works, obviously, for colorized terminal output. As far as I can tell, there is no option to indicate moves in plain text patch format, but that makes sense.
For default behavior, try
git diff --color-moved
The command also takes options, which currently are no, default, plain, zebra and dimmed_zebra (Use git help diff to get the latest options and their descriptions). For example:
git diff --color-moved=zebra