I must confess that I skim read the OP post at first. As indicated by E17phil, unless you have a current version of the installed software or the installation media (evidently not), you're basically screwed with or without the installation / activation code(s) you can cull from Belarc Advisor.
You could of course, (as indicated by realspeed), clone the hard drive as an image with Symantec Ghost or perhaps Acronis True Image - or similar.
This still leaves you with a problem though.
Theoretically, if you have the OS install DVD for the new PC, you could wipe the boot drive and write an image to the drive you ghosted to it, then upgrade to 7 with the install media.
There are likely to be horrendous problems though as not only are you upgrading the OS, you are working with new hardware. The HAL will go nuts and the device manager look a picture.
In all honesty, I think you'd be very lucky to get away with this.
There is an application called the Universal Imaging Utility (UIU) from Binary Research which will - allegedly - allow you to transfer a working system to new hardware; never tried it - and - expect it's expensive.