This same problem happened to me.
My WD portable hard drive wasn't showing up when I plugged into my Mac. The blue light turned on, I could hear it whirring, but it wasn't appearing.
I called WD, and they told me that this sometime happens when the hard drive isn't receiving enough power. (This happens ESPECIALLY WHEN USING MACs)
They promptly sent me a new connecting cord that plugs into both of my USB ports in my MacBook. Problem solved.

Hello. I've just had this very problem (and panicked)... but the solution for me was simple..no need to try and mount through disk utility etc etc (which I tried first). ALL I DID WAS TO TURN MY MAC OFF, UNPLUG THE EXT HARDDRIVE FROM THE WALL & REMOVE ALL WIRES CONNECTED TO TO UNIT, UNPLUG THE MAC FROM THE MAINS, WAIT A FEW SECONDS, PLUG IT BACK IN, PLUG THE HARDRIVE TO THE MAINS AGAIN, REATTACH THE FIREWIRE AND SWITCH ON THE POWER TO ALL. THE BEAUTIFUL LIGHT REAPPEARED AND WAS MOUNTED TO THE DESKTOP. SORTED. TRY THIS BEFORE ANYTHING ELSE!

Allright!!
I dont know if this will work for everyone.
Regarding my previous post i had taken the hard disk out of the enclosure and plugged it directly into the motherboard via sata to no avail.
I then decided to put it back in for enclosure and thought i will try it again.
In device manager i realised the hard disk kept trying to refresh inself and would cut off before it got recognised....playing with the enable/disable option i managed to (somehow) get the drive working(cant explain how yet)
But it worked long enough for me to make my backups.
So i would recomment trying to play around with device manager....might get it to work.
Also its intresting to know that if you try to claim on the warrenty using the serial number on the enclousure you get only one year warrenty but if you open it and enter the serial number printed on the hard disk itself you get a longer warrenty.....
Hope this helps
Good Luck