Thanks to both of you -- that's super helpful! For reference, our flights were booked directly with Qantas: one for domestic Australia travel, one for travel to the US, and another for travel to Ecuador.
number_6: when you canceled online, do you remember if you were you shown how much your refund would be before finalizing the refund request? I'm just afraid that once I hit "Request Refund" it'll all be done automatically/immediately before having a chance to review the amount and before having an option to cancel the request (if there are any potential cancellation fees -- in which case I'd call up and speak to an agent). Thanks again!
In case it's helpful to anyone else: for AA flights cancelled by AA (and booked by me from Australia, in AUD), I was able to do this online @ prefunds.aa.com easily (I originally tried to do this with the US Exec Platinum desk, but they couldn't do it for me because the flights were paid for in AUD). It was "Pending Review" for about 5 biz days before it said that it was credited back to my credit card -- and it then took about 3 biz days to go back to my card.