You don't have to do a clean install.Just plug that HDD in your new computer and then repair Windows using its CD or DVD disk.The REPAIR process will adapt the current Windows OS to your new hardware and it will work like charm.
NOTE: Be sure that the CD or DVD disk with which you are doing the REPAIR process of your Windows OS has the SAME service pack version as the one installed on your HDD!!!If it doesn't,you can always slipstream the service pack.
Cheers!