If you're talking directly from device, you can't, nor video. Unless you want to delete everything, which I don't think you want to you can try the steps below.
I think you can delete music on the Zune HD, though.
(If you want to know how to delete everything from a zune anyways go here: http://support.microsoft.com/kb/926917)
If you have the software try this: (Original Post @ http://answers.Yahoo.com/question/index?qid=20080825153604AA868aU)
1. Connect your Zune device to your PC and start the Zune software if it doesn't start automatically.
2. Click the camera icon near the upper left corner of the window to be sure you're in the "picture" view.
3. In the left column under the name of your Zune, click "All Pictures". The center pane should now list all your pictures.
4. Left-click the first picture listed. Scroll to the bottom of the list and shift-left-click the last picture listed.
(Now every picture from your first to last (all) should be selected.)
5. Right-click any picture within the selected group and click "delete".
OR: (Original post at http://www.zuneboards.com/forums/gen-1/983-how-delete-videos-pictures.html)
I thought I'd post my solution for getting videos and Pictures off the Zune. It frustrated me for about an hour. Finally I figured out the process to get them off. As bad as Videos and Pictures were, the music was worse. Especially given that it auto added just about every media file on my computer.
First! Make sure the directory containing the files isn't being indexed by the Zune software.
Second: Right-click your Zune in the left hand menu. Select "Do not Sync Automatically" You can turn this back on after you finish the process.
Third: Delete the file(s) from the Zune (It will have to be connected to the computer), and the delete the files from your computer.
Fourth: Re-enable "Auto-Sync" if you wish. (You probably want to)
The problem is that usually, the stupid files sync back and forth once you delete them from one location. So it's virtually impossible to keep them away without turning off auto-sync. The reason this doesn't happen with an iPod is because the iPods only sync from the computer to the iPod, not vice-versa. Because it's possible that your Zune got music wirelessly or as a guest account on another computer, Zune software has to take into account that you want to sync that back to your computer.
Make sense? Help anyone?
The funny thing is there is an option that says something like, "Don't sync previously deleted files" and it does it anyway.
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