Please see this note for more on non-x86 hardware.
Intel 386SX/DX/SL, 486SX/DX/SL/SX2/DX2/DX4, Pentium, Pentium Pro, Pentium II, Pentium III (regular and Xeon versions), Pentium 4, and Celeron are all supported.
AMD 386SX/DX, 486SX/DX/DX2/DX4, K5, K6, K6-2, K6-3, and Athlon (all varieties, including MP) are all supported. Older versions of K6 should be avoided as they are buggy. Setting "internal cache" disabled in bios setup can be a workaround. Some early K6-2 300Mhz have problems with the system chips.
The old NexGen processors are also supported.
A few very early AMD 486DX's may hang in some special situations. All current chips should be okay and getting a chip swap for old CPU's should not be a problem.
IDT Winchip C6-PSME2006A processors are supported under Linux.
The Transmeta Crusoe processors are supported.
Linux has built-in FPU emulation if you don't have a math coprocessor.
Linux supports SMP (multiple CPUs) in all 2.x kernels. See the Linux SMP HOWTO for more information.
ULSI Math*Co series has a bug in the FSAVE and FRSTOR instructions that causes problems with all protected mode operating systems. Some older IIT and Cyrix chips may also have this problem.
There are problems with TLB flushing in UMC U5S chips in very old kernels. (1.1.x)