I did exactly what you're doing.
After copying 800 gigs of files and throwing away the dvds, I lost everything I had copied on a Seagate Barracuda 1 TB drive. Non-recoverable. So get 2 drives and back everything up!
Anyway, I installed 3 DVD drives on the same machine, and started them copying them simultaneously to the hard drive. I used Teracopy to make the copies queue one after another, and ejected them by hand when they were all done. I had buttons and scripts programmed to eject the disks but found it was quicker just to press the buttons on the drawers themselves. Teracopy will play a nice chime when the copy is complete.
This works, of course, only if you're just copying files from the disks, and not ripping images.