You need a Motorola 680x0 processor with a programmable memory management unit (PMMU). There is no way to run Linux/m68k without one. This reduces the list of possible processors to 68020+68851, 68030, 68040, 68LC040, and 68060. This list of processors excludes the 68000, 68HC000, 68008, 68010, 68EC020, 68EC030, and 68EC040. It also excludes the CPU32 processors (683x0 series) and the ColdFire processor. Linux/m68k can never run on these processors as they lack a PMMU and an interface for an external one (some 68EC030s do have a functioning PMMU and will run Linux; however, their long-term reliability is questionable since Motorola never tested the PMMU). Consequently, all Linux/m68k software is compiled for 68020 or higher CPUs.
Having said this, there has been an effort to create a Linux port that does not require an MMU; it's called uClinux (or Microcontroller Linux), and apparently does what it does quite well, but it is limited in that it can't support virtual memory or memory protection. uClinux and Linux/m68k binaries are not interchangeable. You can learn more about uClinux at Paul Coene's site.
If you can't figure out whether your computer has the "right" processor built-in, please see the next section of the FAQ.
At the moment, current versions of Linux/m68k require a FPU (floating point unit) to run. While there is some unsupported code that attempts to emulate the FPU (see the section called What is the current status of FPU emulation? in Chapter 7, it is not reliable and its use is highly discouraged. If you have a 68020 or 68030, you need a 68881 or 68882 coprocessor as well; if you have a 68040 or 68060, you need the full version of the CPU (not the EC or LC version). So until the new FPU emulator is released, the following CPUs (listed by their full Motorola part number) are the only CPUs that support Linux/m68k:
MC68020 + MC68851 MMU + separate FPU (MC68881/MC68882)
MC68030 + separate FPU (MC68881/MC68882)
MC68040
MC68060