There are three ways (at least) of expressing motion.
1) Stop the motion - catch something that obviously is moving fast at the most "obviously moving" part of its motion eg a ball at the top of it's bounce or a gymnast in the air. I use this in my sports photography - ball in the air, feet off the ground. It's literal and this is the sort of things newspapers want.
2) Panning - now this is fun! But it can be difficult. You need a slowish shutter speed and you need to move the camera at the same speed as the subject. This way the subject is clear and in focus but you get a directional blur in the direction of movement. The difficult part is matching your pan speed to be exactly that of the subject -it takes practice!
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3) Slow shutter speed to express motion - I find this the most rewarding. Here is an example of how (last year) I tried to show the rippling, swaying effect of the wind on the crops. I didn't quite get the feeling from this image that I was after, but I think it does illustrate the sway and I find it interesting and a good example of using shutter speeds to give a creative interpretation.
If you've some good examples of expressive motion, please comment with a link!