If you have a GOOD PC supply (a good many are not...), the rails will be just as clean as any arcade supply you can find. They're the same internal topology, after all, but PC supplies tend to be built a little cheaper and have higher current ratings. Even if it's a cheapo, it's probably putting out power that's no dirtier than the wart that came with the LCD in the first place. Most LCDs will have their own internal power stuff since they need HV for the backlight and much lower voltages for the rest of it.
PC supplies certainly are "regulated". In fact, they even go so far as to recommend that there be a separate sense wire for the 3.3V rail right down at the motherboard to improve 3.3V stability.
You can do what you want, just make sure that your LCD actually wants 12V! Some want 18, some 10, some 8; they're all over the place. In other words, sure try it (if the voltage is right). It's highly unlikely to break anything. If weirdness does occur, try a dedicated power supply (like the brick that came with it).