Hi ewhelhalu,
I just figured out a way to do it, but it's a little weird and complex, and I have no idea what the consequences for editing this .plist string might be. I'm thinking it'll be OK, but you never know what might happen when you screw around with Apple's intentions.
So if you have the Developer Tools on your install of OS X, you can use the Property List Editor to edit the Dock files. (If you don't have this program, you can use a program like [Pacifist|http://www.charlessoft.com> to selectively install stuff from your Leopard disk).
The file you're interested in is within Macintosh HD > Users > Your Username > Library > Preferences, and it's called com.apple.dock.plist. Open that up with Property List Editor, and open the arrow drop-down next to "Root." The lines you're looking for are "persistent-apps" (the left side of your Dock) and "persistent-others" (the right side). If you open the drop-down arrows, you'll see there are numbers below each. The numbers correspond to your Dock items, from left to right.
If you click on drop-arrows by the numbers, a sample hierarchy looks like this: 0 > tile-data > file-label. It's the "file-label" string that you want to edit. Just make it ("it" being the name of the icon under the "value" column) blank, save the file, go into Terminal and type "killall Dock" without the quotes, and your Dock icon should appear without a name. You'll have to do this for all of them.
I should insert the usual caveats here about backing up beforehand and making sure that you have a copy of the .plist file before you start tweaking it so that you can replace it if things go wrong. And perhaps someone with more knowledge of Terminal or third-party apps than I have will come along and suggest a simpler solution. But that's the only one I have. 🙂
Hope that helps!
—Hazy