These instructions are for Thief version 1.37 (Thief Gold) only; click here to find out how to determine your version.
Download ThiefGrck.zip.
Unzip ThiefGrck.zip to the folder where Thief is installed (make sure THIEF.EXE is in that folder).
Double-click THIEFGNOCD.exe.

Click Path and let it rip!

Voila! You can now run Thief: The Dark Project without your CD. I had to run this twice for it to work, I'm not sure if that had anything to do with me changing the processor affinity before hand. If you get the dialog prompting for a CD, try clicking Cancel a few times.