Actually, the Athlon XP incompatibility isn't a problem with BeOS. It's a problem with the Athlon XP.
Here's the deal:
The Palomino core used in the Athlon XP includes a full implementation of Intel's SSE instruction set. So far, no problem--BeOS fully supports SSE. But, AMD did something pretty weird. In their AMD's implementation of SSE, they force the Athlon to report that it's *not* an Athlon but an Intel chip! In effect, with AMD's SSE enabled, the Athlon misrepresents itself to the system as an Intel chip. When BeOS starts to load, it asks the system about the processor, BeOS is told that it's an Intel chip. When the BeOS kernal, thinking it's talking to an Intel chip, sends instructions to the CPU, it crashes the system. The Athlon XP tells BeOS it's an Intel chip and when BeOS treats it like an Intel chip, it crashes.
Disabling the AMD's wacky implementation of SSE solves the problem. This can be done at the motherboard firmware level.
At least one motherboard manufacturer provides boards that allow the AMD SSE to be disabled. All that's required for a motherboard manufacturer to provide this feature is to update the BIOS. Hopefully, Tyan, Asus, and the rest will follow suit and add this option.
I have no idea why AMD chose to do this with their SSE implementation. Perhaps they had a compelling reason to do this, but it seems completely insane to me. I'd still rather have an Athlon XP than anything from Intel, though.
If you have an Athlon XP system and can't boot BeOS, contact your motherboard manufacturer and ask them for a BIOS update so you can turn off SSE.
As for the PIII, both the PIII and PIII Xeon are fully compatible with BeOS. They're both listed on Be's BeOS hardware compatibility list and I've heard and read of many BeOS users with PIII-based systems. I've never tested this configuration personally, but I wouldn't expect BeOS to have a problem with the PIII. Besides, the PIII is essentially a souped up PII, really. The architecture is hardly different.
As far as any incompatibility between the BeOS and the new Intel Pentium 4 processors, I have no idea. The new Intel Pentium 4 is... well... I don't mean to offend any loyal Intel fans, but it's a tremendous dissappointment. It's actually LESS POWERFUL than the PIII at the same clock speed! (The Pentium 4 is roughly 33% less efficient as the Pentium III.) In other words, a 1GHz PIII is roughly equivalent to a 1.5GHz Pentium 4 processing the same code! Isn't that nuts?