Okay, the title is slightly misleading ..

You can not move layers without touching them, but you can move them from the layers palette without clicking on them using the following keyboard shortcuts:

To move a layer down to the layers palette, press: Command-left bracket (PC: Ctrl-left bracket)
To move a layer to the palette of layers, press: Command-right bracket (PC: Ctrl-right bracket)

Now, if you want to move a layer within the image without using your mouse, select the Move tool and use the arrow keys on your keyboard.

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