First thing when diagnosing a desktop computer problem, is check the Power Supply.
A Power Supply with a weak voltage power rail, can also emulate software problems.
(Emulate software problems; The bad hardware (Power Supply), can also make it seem like it is a software problem. It isn't. It's a hardware problem -> Power Supply )
1) Is your computer dirty inside? This will make the computer not work, and also 'kills' Power Supply's.
They overheat, and components inside the Power Supply fail.
1) If ALL of the LED lights were on at once, they would use less than 1 Watt of power.
2) EACH fan uses 2 to 3 Watts of power.
3) A typical Processor can use 51 to 125 Watts of power. Just depends on what Processor it is.
LGA 775 processor socket, and an Intel G31 motherboard chipset,http://en.wikipedia.org/wiki/List_of_intel_chipsets#Core_2_chipsets
Scroll down to G31 in the left column.
Also go back to the Gigabyte Support page.
Click on the Support & Downloads tab.
(Overview / Specification / Support & Downloads / FAQ / etc)
Click on the blue - CPU Support List tab.
(CPU Support List / Memory Support List / Accessory )
Your motherboard can support Processors that use 35 Watts all the way up to 135 Watts.
If you have a Power Supply with a weak voltage power rail; you may see LED lights light, and fans spin; but there will NOT be enough power to turn the Processor ON.
Do you have a multimeter to test the 3 main voltages?
Do you have a KNOWN to be good, Compatible power supply; to use for a temporary test unit?
When working on your computer FOLLOW Anti-Static Precautions.
If you are not aware of what Anti-Static Precautions are, please post back in a Comment.
DO NOT work on your computer until you do.
Remove all Ram Memory modules. ('Stick')
Clean the gold plated contact pins with a pencil eraser.
If using an eraser on the end of a pencil; and not a solid eraser; refrain from letting the metal band of the pencil; touch the gold plated contact pins.
Rub across all gold plated contact pins, and up, and down. BOTH sides.
When finished use air to remove the eraser dust.
You can use a can of compressed air for computers, or air pressure from your mouth will be sufficient.
Handle the Ram Memory module by the BODY.
The Body is everywhere on the Ram Memory module, EXCEPT the gold plated contact pins.
The Body is coated with a see-through protective plastic,http://en.wikipedia.org/wiki/File:Generic_DDR_Memory_%28Xytram%29.jpg
Reinstall the Ram Memory module/s. Plug the computer back into power. WAIT 1 minute, turn the computer on.
For additional questions please post in a Comment.