Generally you only 'need' to format the drive if there is a major version OS change going on or if the hard drive controller is drastically different(or if you have hardware RAID). The reason people often find it easier to reformat is because operating systems do not cope well with changing key drivers on the fly. In the case of Windows OS's the three biggest problem drivers are the Motherboard chipset drivers, the storage drivers for the Boot disk controller, and the Graphics card driver..
Unless you plan your change so that those three controllers are unchanged you will probably get a blue screen if you just connect your existing drive to a new motherboard. Most of the time a different CPU means different chipsets too so a straigh swap almost never happens.
The lowest cash cost method is to reformat the boot drive and reinstall everything, but it often involves a large investment of time and effort. If you upgrade the OS as a part of the process you may also find that a lot of your older software is no longer compatible and be forced to upgrade more stuff than you expected inflating the cost.
If you are using Win7 and not planning on changing the version/edition of Windows at all you may be able to usethe built in repair install feature to do this.. I would definitely recommend a backup first though.. I have never tried this method before for a Mobo swap but have used it in VM repair. The following link outlines the details.
http://www.sevenforums.com/tutorials/3413-repair-install.html
Having said all that.. What I usually advise people to do when upgrading MOBOs or even entire systems is to do one of two things.
1.) take the opportunity to upgrade everything.. Hard drive, OS, and software. (essentially start over)
2.) use an imaging tool that can handle system driver changes for you along with an external drive.
The only product I have used personally for this is Acronis True Image with the Plus Pack. You would essentially buy the software. make an image backup of your existing system to an external drive, install the new components, use the imaging softwares boot disk to restore everything onto the new setup. There are probably similar products from the other imaging software vendors, but I have not had any experience with them. There are also manual ways to do the driver swapping, but they are a MAJOR pain especially if you are switching from unicore to multicore or across CPU architectures. I would never recommend the manual method to anyone except a hardcore geek, I did it once for a client, it sucks.
Note that in order to do this you will need your original OS and driver media and it is simplest for hardware that the OS supports natively, complex disk channels make it harder because ou may have to 'slipstream' the drivers first. (If you don't know what that is you probably don't want to try it)
Unfortunately as far as I know free cloning software would lack the driver swapping elements..