Scouring pads and other abrasive scrubbers can ruin the nonstick plates on a panini press, but water is gentle and effective. This technique is especially handy for panini grills that don't have removable plates.
With the panini press unplugged but still hot, we take a hot water-soaked cloth or sponge and gently wipe the plates. (If you're worried about your fingers, you can grip the cloth or sponge with tongs or chopsticks.) For more stubborn residue, we pour a little bit of hot water onto the plate, close the grill, and let it sit for a minute. These methods work best if cleaning is done right away, before the residue has time to settle into the grooves.
As always, be sure to check your manufacturer's instructions for more complete cleaning advice.