Start by recognizing the the information on your HDD is fully readable by your new system - it is not changed. That is, AS LONG AS you were NOT using a RAID drive array, but since you refer to ONE drive unit I expect you were not in RAID anything. There certainly could be an issue IF your original drive was set with a password to protect it from unauthorized access to the My Documents folder. Did you set that up? Then I would suppose you already know the password.
The root of the problem is that the fully readable info happens to have a few things wrong in the context of a new mobo. When Windows of whatever version is first installed, one of the important things done is to load with it drivers (software) for all the hardware devices in the system. Now your motherboard has many hardware devices that need drivers, and those all are on your old HDD suited to your old mobo. If you simply transplant that HDD onto a new mobo, odds are there will be some, possibly MANY devices that are different, and the wrong drivers will be loaded from the old HDD.
One way this can be solved, often successfully, is a Repair Install. To do this you need the original Install CD used to install your Windows, and an optical drive to run it. Having installed the old drive in the new mobo, you boot from the Windows Install CD in the optical drive. BUT do NOT do a normal Install. Look for the option of a Repair Install and do that. It will look at the hardware devices currently found and the drivers already installed on the HDD, and try to correct all the mismatches. This at least will get you running in most cases. Then to be more secure, you should really download from the mobo maker's website all of the latest drivers for your mobo and update every one of them, just in case the older drivers on the Windows Install CD are out of date.
This process solves the problem of being able to load and run Windows properly, with all the correct device drivers, from the HDD. When that is done, all the rest of the HDD's info is all there, ready to use. If you had password protection on folders, you'll need to use that to get into them.
There are times when this process does NOT work - the driver mismatch problem is just too complex. In those cases the only good solution is to re-install the entire Windows OS. Now, doing that on your old drive will most certainly wipe out its old files, DON'T DO IT! That is where fihart's solution is best. Buy and install a new drive and do NOT install your old one. Install Windows, whatever version, to this brand new empty drive. Once the machine is running this way, THEN you install the old drive as a second unit, and it will show up in your new machine as just another drive full of files you can use. You then copy all your old data to folders on the new drive. Eventually, once you are SURE you have ALL old files copied that you want, you can wipe the old drive clean and continue using it as a second data storage device.
The third option, certainly more complex and therefore somewhat risky, is to try to clear out as much empty space as you can on your existing old drive in the old machine. Then you use some third-party software utility to shrink the Partition now called the C: drive, leaving some Unallocated Space on that HDD unit, and convert that to a new second Partition. If you can make this large enough, you can then use it as your "new drive". You replace your mobo and PSU, etc, install the old HDD unit, and install Windows on this new second Partition. Now your "new drive" and your "old drive" happen to be two Partitions on the same physical HDD unit. This process can work but it does risk problems that make the drive unreadable, so you lose everything unless you have a good backup first. That is why buying and installing a new physical HDD unit, and installing Windows to that, is safer. In that scenario all you are doing with the old HDD unit is installing it to a running system AFTER it is all working well, which has much less risk. But, even easier if it works (and it often does) is the Repair Install procedure that can "update" your existing Windows installation on the old HDD to work in a new environment.