All attempts to use Norm's script failed for me. I have no idea why, so I tried something else. Even with Automator, it did not work, so I ended up with this little AppleScript:
tell application "Terminal"
activate
do script ¬
"diskutil unmount \"/Volumes/volumename to unmount\"" in ¬
window frontmost
delay 5
quit
end tell
It activates Terminal, invokes Apple's command line tool diskutil in the frontmost window of the Terminal app, unmounts the Volume with the name Volumename to unmount, waits five seconds (until the unmount progress is finished, otherwise a sheet opens and asks for user interaction, which is very inconvenient), and then quits the Terminal.
The script works fine for users with administrator accounts, but standard accounts do have to enter an admin name and admin password. So it's not an all-rounder. If there is anyone who has an idea how to unmount volumes in Tiger while logging in as a standard user, please help me.

