Argthtjtr

In 1981, Microsoft released Xenix which could run on the x86, the 68000, and others. As you know, Microsoft targets microcomputers above other markets; that was especially true in the 80s. So this was perhaps the first unix, or at least among the first commercial unices, to be intended to run on high-end micros.

But maybe, the term "micro" isn't so useful. After all, Version 6 UNIX from 1975 was written in C and ran blithely on the PDP-11. The PDP-11 architecture was later implemented on a microchip; A J-11 or something like that would have run V6 just as happily. And in Soviet Russia, V6 was indeed modified to run on local PDP-11-compatible microcomputers. It's arguable whether MNOS and V6 are the same UNIX; if they are, then it predates Microsoft's offering by around five years.

BYTE magazine, October 1983 talks about early unices on microcomputers

It all comes down to the definition of micro. After all, already the PDP-11/03, aka LSI-11, of 1975, is based on a micro chipset. That's less than a year after Unix became known outside AT&T. An 'official' AT&T port to the LSI-11 was published in 1978. Not much later, the PDP-11/24 (1979) used the F11 chips follow us where J11 and T11 (single chip PDP). Clearly micros (*1).

When it comes to more common (*2) CPUs, Intel's 8086 got its first port, by AT&T in 1978, shortly followed by Microsoft's XENIX, based on the AT&T port (*3).

*1 - Or do the case form make the difference? Then I must point to some 6502, Z80 or even 2650 based systems in 19"

*2 - not that I would call the LSI-11 uncommon

*3 - Sans the custom MMU that is.

According to the chronology of personal computers, the first micro-computer running Unix was “the Onyx C8002 microcomputer. It features a Zilog Z8000 microprocessor, 256 kB RAM, tape backup, hard disk, serial ports for eight users, and running UNIX, for US$20,000.” It was introduced in June 1980. (The quote is inaccurate in at least one respect: the C8002, as its name suggests, used a Zilog Z8002 CPU — thanks Raffzahn!)

This was followed shortly by Xenix, Microsoft’s licensed port of Unix, which first shipped in January 1981 on a Z8001-based Central Data Corporation system (not to be confused with the more famous Control Data Corporation). 8086 systems running Xenix started shipping in 1982. Xenix was also available on PDP-11 computers (see the OEM list for details), and there were micro-computer-class PDP-11 systems, including DEC’s own Professional range which was available with 2.9BSD.

With the other answers, it is obvious the Sun-1 was not the first. However, due to Sun Microsystem's relevance in the microcomputing world, I think this May, 1982 entry is worth mentioning here. The system was based on a 68000 CPU along with a Sun-designed MMU. (Apparently the Motorola MMU at the time was not sufficiently reliable.)

1. 
   Sun-1, 1982, 68000, Sun MMU


2. 
   Sun-2, 1983, 68010, Sun MMU


3. 
   Sun-3, 1985, 68020, Sun MMU, Motorola fp math coprocessor


4. 
   Sun-3x, 1989, 68030, inherent MMU, Motorola fp math, system coincident with SparcStation 1
   

No more 680x0 models past that, and later Sun models would either use the Sparc or x86 CPUs (both 32-bit and later 64-bit).

In the early 1980s, Torch Unix was available for the BBC Micro.

The catch was that you had to have the Torch 68000 second processor and hard drive. This meant that it fell between two stools. It was far too expensive for use by the usual BBC Micro hobbyists and office/professional users found it too unusual to compete with more mainstream offerings from other manufacturers. So it soon died out.

I did use this at work for a short time. I remember it had to be installed onto the hard drive from 50 5.25" floppies, this took me pretty much an entire working day.