No not really.
When you buy the gigabyte motherboard, it will be rated with a certain socket type. This is the type of cpu it can handle. It will also be rated with a certain amount and type of PCI slots.
I.E. LGA 775 for Core 2 Duo & 1x PCI Express 2.0 x16 & 2x PCI Express x1
When you buy a graphics card, it will require a certain amount of power and a certain PCI slot.
I.E. 400 Watts and a PCI Express 2.0 x16
When you buy a hard drive, it will require a certain type of connecter for both data and power.
I.E. Sata & X number of pins
When you buy a CPU it will have a certain type of socket, it will only fit in motherboards which support this socket type.
I.E. LGA 775
There are many other factors as well including:
Motherboard size (Micro ATX, ATX, etc)
Case Size ("")
Memory Type and Size (nGB, DDR2-3, Dual/Tri Channel, XXXMhz)
Sound Card Socket (PCI Express, etc)
In other words, yes you can buy a gigabyte motherboard and stick an amd cpu in it, but the gigabyte motherboard has to have to correct socket for that AMD CPU (AM2, AM2+, AM3)
I highly recommend finding/buying a beginner's guide to custom pc building.