I have the Gigabyte Gaming 7 motherboard, and I looked at the pic for the gaming 3. This style motherboard have the PCIe 16x slot in the second position. Hence, there is room for a wide PSU heatsink like the Noctua NH-15. For motherboards that cater to HEDT chips, they tend to have the 16xPCIe card in the first slot. That is why Noctua sells the NH-D15S. The fin stack has been moved over to accommodate having cards in the first slot.
As for tallness of the heatsink, the rule of thumb is this: if you have a 120mm exhaust fan in back, you can fit a full-height heatsink like the Noctuas under the sidewall of your case. With a 235mm case, you might be able to put a fan inside that sidewall without interfering with your heatsink. If you have a 140mm exhaust fan you should be able to fit an extra-tall heatsink in there like Thermalright's original Archon and Silver Arrow.
I recommend removing the rear "grill" of your case. You can cover the cut sides with trimmed plastic spines that make up cheap report covers. That "grill" really restricts the outgoing airflow -- so much so that you don't need a case exhaust fan if you remove the "grill." Even better, if you have a pull fan on your heatsink (on a tandem tower that means a third fan) the pull fan will entrain case air and serve as an active exhaust that way. To see a hilarious example of that, look
here.
Most of all, have fun. You can hardly screw things up. Lots of fail-safes these days.
