This is a common problem on Piper gears. I recommend that you check your nose gear torque link bushings first. Have someone push down on the tail so that the oleo is half extended while you go and wiggle the torque links. I am certain that you will find a lot of play. The bushings are cheap & also be sure to have the bolts replaced.
One other thing to check is the steering bungees that go through the firewall. One may be weak & I have found this before.
Any play or bad bungees will definately cause nose wheel shimmy as the trailing edge of the nose wheel pant has an arm to it causing an inbalance so to speak.
Last but not least check that you are running the correct tire pressure.
I have worked on many PA 28s in a flight school so I hope that my advice helps.