The procedure, which renames the arrow-image file AliasBadgeIcon.icns so that the System won't find it, affects all users and accounts and requires administrator privileges. Open Terminal and type or copy/paste:
cd /System/Library/CoreServices/CoreTypes.bundle/Contents/Resources
Press Return and then type or copy/paste:
sudo mv AliasBadgeIcon.icns AliasBadgeIcon_off.icns
Press Return again, type your administrator's password at the prompt, and you're done. The arrows will be gone after your next restart.
To undo this, repeat the steps above, but swap the positions of the filenames on the second command line:
sudo mv AliasBadgeIcon_off.icns AliasBadgeIcon.icns
All thanks and credit to the original posters who figured this out. [crarko adds: The correct authorship has been established, and thank to JimAkin for the original hint. Apologies for the confusion.]
[crarko adds: I tested this in 10.6.4, and it works in Snow Leopard as well.]

