From what you describe ...
Either the drive letters changed or the drives can't be seen by the computer as the shortcut is simply to the drive itself and if double clicked would open that drive in a Windows Explorer window IF THE DRIVE IS RECOGNIZED AND ASSIGNED THE DRIVE LETTER by the OS.
Are A, C, D, E (& J (I am betting that the (& might be drive I) the drive letters you normally see in Windows Explorer (or My Computer)?
Right click on My Computer and select Manage. Now select Disk Management (if you don't see it expand Storage) and see if the drives show up and what Disk Management says about each. If they are there but the drive letters have simply changed you can use Disk Management to assign the proper drive letter to them again.
If your computer can't "see" the drives make sure they are turned on and if they are do as Kees suggested to discover if the drives are visible to another computer.
If not I wouldn't toss them right away, I would try taking them out of their enclosure and getting a converter to interface their connector (IDE or SATA) to USB unless maybe they are firewire or SCSI drives.