Everybody hears how the RIAA and the MPAA can track your IP and ask your ISP for your information.

My question is: how do ISPs know which customer has what IP?

My theory is that they check their DHCP server logs to see which IPs are issued to whom.

But, I could just make linux claim another IP and do my leeching from there instead, and dispose of it when I don't need it anymore. Would that make me hidden from them?

I know that the MAC address could link it to me, but do routers keep track of the MAC address and which IPs they used?

I'm relating these questions to a cable ISP (optimum online).