We've had a similar ban here in South Australia for a number of years now. It has worked quite well - one either remembers to bring a bag when grocery shopping, or buys a reusable one then.
From link: One noticeable success in Ireland's environmental track record was the introduction of a plastic bag levy in 2002, the first country in the world to do so. All consumers were required to pay 15c for a plastic bag; this led to an immediate decrease of over 90% in the amount of plastic bags in circulation. From 328 bags per inhabitant per year when the levy was introduced, usage fell to 21 bags per capita.
http://www.smithsonianmag.com/science-nature/corn-plastic-to...
Only cottoned onto this personally, after noticing the "compostable" bags I'd been using to hold vegetable scraps... weren't compositing in the home compost bin even many months later (nearly a year now). :/