For 4G sticks you need to the Gigabyte site at your mobo page, check the memory support list and see if you find a 4GB RAM with the same voltage, frequency and timings as your current RAM (note down the RAM codes and look them up on the net). Probably you won't find much in there so go to newegg's mobo page and scan the comments, note down the 3 x 4GB RAM they say they use with the system and check those for voltages and timings too. Do the same on the memory pages, select 1600 DDR3 RAM your voltage and 12GB (3 x 4GB) and see if there's any RAM with the same specifications as yours.
I know it's painful but all the mobos have issues with 4GB sticks, many are not compatible.
Does this RAM fit yours?http://www.newegg.com/Product/Product.aspx?Item=N82E16820231406
Make a new thread in here named "Need 12GB (3x4) 1600, "x" voltage, "y" timings for my GA-X58A-UD3R mobo" or something, hopefully somebody in here with that mobo can give you some pointers.
You need to get a 12GB kit right off the bat not one or two sticks because they're tested to work together in triple channel.