As siren already pointed out, there will be a compression wave (longitudinal wave) travelling down the rod when you move it. The velocity at which this happens (and thus the limiting speed for communication) will be the speed of sound in the material the rod is made from. This will definitely be a lot slower than the speed of light; for instance the speed of sound in iron and aluminum are approximately 5100 m/s.

For example, hit a nail with a hammer; it will still take a fraction of a second before the tip "knows" that it should move.