Today I got another call, this time with my own information on the caller ID. It was the "lower the interest rate on your credit card" scam. Usually you have to "press 1" to talk to someone, but in this case a guy was already on the line. I took Gelston's advice, said "just a minute", put the phone down, and walked away.
20 minutes later, the guy was still waiting on the line. I came back an hour later, and he had given up after 55 minutes (the phone records the length of each call). These idiots certainly are determined.
It turns out that my call blocking device won't let me block my own number, so that's probably why these clowns are using that technique. I'll have to research the issue.