I had the same problem when I lived in Melb. Planted grocery-bought pieces in good friable potting mix in a black pot in spring/early summer, placed on concrete up against a brick wall in a sunny spot, so the pot got radiated heat as well as lots of direct sunlight. To no avail, though. Can't remember if they failed to germinate entirely or just started germinating and then died. (Mind you, I managed to get some boabs started from seed, but despite my best efforts the seedlings never made it thru the winter.) However, I remember my mother saying recently (when the price of fresh ginger hit the $20/kg mark) that one of the Vietnamese green-grocer staff in Melb told her that he grew ginger himself, so it must be possible.

Lack of sustained heat is a problem when growing tropicals in Melb, and I remember some 18 degree days when I was there in Jan this year ...

I wonder whether the grocery ginger is sprayed with some chemicals to prevent re-propagation. I know that Elephant garlic imported from China is sprayed with chemicals, the effect of which is to prevent further shooting of offsets.

Dunno about the spraying. I've bought fresh ginger from supermarkets in both Melb and Bris; no luck in Melb, but grew like crazy in Bris. It's possible that the Melb-bought ginger just happened to be sprayed and the Bris-bought one wasn't; or it could be that they were both sprayed but the continual warmth in Bris allowed the rhizome to sprout anyway. (I just wish I could remember whether the Melb-planted ginger managed to sprout at all ...)

There is some good advice in the Ginger Forum here at garden web US on growing edible/ grocery store bought ginger. I am too trying to grow some now. The article by Tim Chapman helped answer alot of my questions.
Good Luck

Ginger is a tropical plant. Cool nights will stop the rhizomes dead in their tracks. Dead being the operative word. I am having great success growing ginger in Geelong in a glass house. They are in pots, in semi shade, in rich well drained potting mix and compost. But, they haven't seen the winter yet! This year I will plant again in August and am confident of a harvest in April - I hope....