HTML5 Beta YouTube Video Download
Or just drag the YouTube Unlocker Bookmarklet onto your bookmarks bar and you should be good to go. This has only been tested in chrome.
Flash YouTube Video Download
If you still use the old Flash way of viewing videos, they have also improved their algorithm so that videos are deleted as soon as they are opened in /tmp/. This means that you can no longer trivially copy them out. Fortunatly the file still exists on linux systems, and it's not impossible to copy them out of the tmp folder. Check out the following command line (which I pasted into a button on my panel) that opens the video in VLC. Similar could be used to copy the file to your home folder, or whatever you wanted to do with it.
On a technical level, this command looks up the flashplayer executable and finds open file handles (including deleted files) to Flash files in your tmp folder. The string is then rewritten into something that can be passed into VLC. If you run this script (or hit the button on your panel with this script in it), VLC will open with any currently open videos, making watching the video easy and nice again.