I've had to cancel the Times a couple of times over the years. Telling them that I lost my job only got one offer to extend at a discounted rate. But what really worked well was this:

NYT: "Can I ask why you want to cancel your subscription?"

Me: "No."

It's not on their script, so the operator just took care of it immediately.

On the plus side, at least you can speak to a human being at the Times pretty quickly. Good luck getting through to anyone at the Albuquerque Journal in under three days ("It looks like you're calling from a cell phone. Let's continue this conversation by text!").