Everyone I know just pays for the new thicker bags. It's $0.30 for three bags on a $100 grocery bill.
The real annoyance comes from people not knowing exactly how many bags they need, and usually guessing too low (not wanting to be wasteful), so the baggers take longer trying to figure out how to squish everything in the bags... and sometimes the customer has to pay an extra $0.10 if they need another bag. Which gets them to dig around for a dime. This probably slows down checkout lines by a significant amount (10% or so?).