My workaround to this has been to email the company telling them I want to cancel. Once I either don't get a reply, or get a reply saying "just call us and we'll cancel!", I dispute the next charge with American Express and have the email record of trying to cancel. I believe they also offer a "stop allowing charges by this merchant" feature that cuts off future charges.
> I believe they also offer a "stop allowing charges by this merchant" feature If they have this it's another reason to use them for automatic billing. I have tried to do this with a VISA card and they said they cannot do it; the only way to prevent future charges would be to close that account entirely and even then I might still get billed for some period of time.
But to add - I discourage you from using chargeback as a feature to stop future charges. Most banks will report it to your credit bureau - you won't see it in form of points being withheld BUT it might be adverse for you when you try to get a loan, etc. My mother disputed way too many things (memory troubles at her age) and they did not renew her CC after expiration date and MasterCard told her she is not eligible for card with her excessive CB ratio.