The simple answer is no, there isn't an easy way to do it. Unfortunately the 360 will only allow MS certified devices to work on it. What you would have to do is take an exisitng 360 controller and hack it to work with your current stick. If the 360 stick you hate has a common ground PCB, and your Xbox stick uses standard microswitched buttons and stick, you would just have to rip all the Xbox stuff out of the old stick and run wires from the buttons to the PCB out of the 360 stick.
It would probably be easier to either mod the 360 stick into something you like, or just get a new stick.