It can be argued that as much of a GPU hog Doom 3 is, it is just as demanding on your CPU. The recipe to success is much simpler on the CPU side however: Doom 3 needs cache and lots of it.
On the Pentium 4 side of things, if you've got anything with less than 512KB of cache it's time for you to upgrade. Prescott owners will be happy that their chips are finally faster than Northwood in something thanks to larger caches.
AMD owners have much more of a reason to rejoice: the Athlon 64 runs Doom perfectly. It's almost as if the game was built to run best on an Athlon 64; maybe AMD should invest some marketing dollars in their own "The way it's meant to be played" campaign. And to make things even better, you don't even have to have the fastest Athlon 64 to get great performance, even the meager 3000+ manages to offer performance equal to that of Intel's Extreme Edition Pentium 4 at a much lower cost. The key to AMD's success is the on-die memory controller; with lower latency memory accesses than the competing Intel solutions, Doom 3 sees system memory as one big cache and drives performance up considerably. It is also the on-die memory controller that makes cache size less of an issue on the Athlon 64, while too small of a cache seems to make or break performance with the Pentium 4.
The Athlon XP is much less impressive under Doom 3 thanks to its lack of an on-die memory controller; unless you have a Barton based Athlon XP, it may be time to bite the bullet and upgrade to an Athlon 64. That being said, the entry level Sempron 3100+ offers very competitive performance at a price point that's low enough to make the transition to a Socket-754 platform relatively painless.
If you are lucky enough to own any of the GeForce 6 series cards and play at resolutions lower than 1280x1024 rest assured that money spent on a faster CPU is money well spent. If you happen to have a slower card, something along the lines of a Radeon 9800 Pro or even a regular X800, your system is far less CPU bound and you may want to go with a more middle-of-the-road CPU in order to maximize performance without spending needlessly.
In the end, the winner of the final battle is clear: the AMD Athlon 64 is the processor for Doom 3.