Not sure if this wasn't possible before but if you select "unsubscribe" from the emails you receive, this leads you to a page where you can manage your subscription, including removing a city, without having to close the account (which you don't want to do if you have unused Groupons!)
I actually tried changing my city on my account and am still getting emails/deals only from my previous city. This was after several attempts at updating my city. I'll try unsubscribing.
scroll all the way down and choose browse city, choose your city and it will become your default city, worked for me. btw how do you unsubscribe groupon as a whole not just the emails?