The link in pap1405's post solved it for me. Here's what I did...
1) Connected my iPad to my laptop
2) Because iTunes wasn't recognizing it, I updated the driver for my iPad using the steps for a Windows PC here - Manually Update the Apple Mobile Device USB driver (near the bottom of the page)
3) Disconnected my iPad from my laptop
4) Shut down my iPad via holding down the Sleep/Wake button and using the slider to power it down
5) Opened iTunes
6) Connected my iPad to my laptop
7) Held down the Sleep/Wake and Home buttons for a good 20-30 seconds. During that time the Apple logo appeared on the screen like it was starting up. However, I kept both buttons pressed through that until the iPad shut off and came back on again. That's when the Recovery Mode popup showed in iTunes (rather than the one that asks if you want to allow the device to connect to your computer... then tells you to respond on your device... but you can't because your device is locked).
I hope this helps someone.