Intel/AMD/Cyrix 386SX/DX/SL/DXL/SLC, 486SX/DX/SL/SX2/DX2/DX4 are supported. Intel Pentium, Pentium Pro and Pentium II (basically it's a Pentium Pro with MMX) also work. AMD K5 and K6 work good, although older versions of K6 should be avoided as they are buggy. Setting "internal cache" disabled in bios setup can be a workaround.
Also IDT Winchip C6-PSME2006A processors are supported under Linux ( http://www.winchip.com).
Linux has built-in FPU emulation if you don't have a math coprocessor.
Experimental SMP (multiple CPU) support is included in kernel 1.3.31 and newer. Check the Linux/SMP Project page for details and updates.
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.
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)