Microsoft and other companies have spent a good deal of time tackling this problem for their products (for example, Windows Genuine Advantage). But here's a quick way to get a hardware footprint of a system from Linux, and your live CD will probably have these tools built-in.
Start with the "lspci" tool. This tool shows, in human-readable text, the hardware visible on your PCI bus. For example, it will list out your network card, video card, sound card, and some motherboard components.
Now, you could use "lspci", but you're probably looking for something a little more concise. I suggest hashing this output into an MD5. To do so, type:
lspci | md5sum
This will redirect the "lspci" command into "md5sum" which will return an MD5 hash that looks something like this:
While this is certainly a less secure way to handle hardware fingerprinting, it's a quick and easy way to get the job done. You should know that it's quite likely that if you swap out a component for another of the same model and revision (say, you bought two NVidia GeForce 7800GTS cards at the same store at the same time), you would probably get the same hash. So, if that's a concern, this may not be for you. But if you and I both ran this command, we would get wildly different output.
Hope that helps!