Unable to get T3i to shoot video in manual mode
Ok, I figured it out. It's a bit quirky little trick.

You do not turn the dial to M but turn it to the little picture that shows the video camera.

Now, though you'll be able to set the shutter speed and aperture, you won't be able to change the ISO.

To be able to change the ISO, click the Menu button to the top-left of the LCD screen, then from the menu, select Manual as the value for the Movie Exposure setting.

Now, you can change the ISO as well.

I figured it out by looking here and here.