The biggest problem will be activating windows again. XP keys are tied to motherboards. It is possible to call microsoft support and get them to issue a new key if you're replacing your motherboard because it died or something. An exception to this is if you bought a copy of Windows XP retail. The retail box is easier to move from PC to PC.
Second problem will be drivers because the new computer will most likely need different drivers than the old computer.
Unless the hardware is very similar, Windows XP is very bad at being transferred to a different computer, you nearly always have to reinstall the operating system. Still no harm in trying you may be lucky.
I've done this a couple of times and never had a problem activating windows.
Drivers are an issue.
1. XP does not support SATA drives until service pack 3, so make sure that you are current on all XP service via windows update.
2. XP was created a long time ago. The parts in your new PC were probably created after XP shipped, so XP does not have the correct drivers. These include MB, chipset, etc. If you bought the MB in the new PC then the MB maker will have a set of drivers you can download. If the new PC was bought pre-assembled (e.g. Dell, HP) then you are going to have to find and download drivers youself. If you can get soemthing runnign then windows update will complete the process of finding you the right drivers. If you cannot find enough drivers to get the system runnign then windows cannot run so windows update can't get them for you.
Good Luck. Post results when you try it.
p.s. 31-bit XP will only use 3.4GB of the memory on your new PC.
p.p.s. Win8 is avail for free (timelocked) download. Then you'd need to purchase win8 for $40 upgrade over XP.
edit: forgot one. The worse driver problem of all is if you switch from an ATI/AMD viceo card to nvidia or visa versa. Before you move the disk uninstall the video driver and get back to the standard VGA vide driver. Taking the 5 mins to do this will save you a ton of problems.