I recently had this problem, but mainly with the hot water. Some research turned up 2 things:
1) Some people claimed that the anode rods in the hot water tank reacted to cause the sulfur smell. My type of tank had no easy way to change the rod, so I did nothing here.
2) I also suspected bacteria; I keep my tank at 110-deg because that is the perfect temp for me coming out of the spigot. Anyway, several articles claimed that the temp had to be a min of 122-deg F to prevent dangerous bacteria. I know that the food danger zone is 40-140, but keeping the hot water at 140 (125+) is a serious scalding danger.
Anyway, I set it to about 122-123 and within a week or so the smell is pretty much gone from the hot water.
P.S. If you go the bleach route, get the pure unscented kind. Also, don't forget that most of the bleach is sold concentrated these days, so 1 gallon of 10 years ago probably comes in a much smaller container these days.