i386, i486, i586, i686 will all perfectly run on any modern processor. The difference which processor it was compiled for. i386 was compiled for the old 386 machines, so any computer with a 386 or higher processor will run it. i686 was compiled for Athlons, P4s and will not run on an old 386 or 486 machine.
Even the newest processors are called 386 architecture, because they inhereted the architecture from 386 processors, just added speed and funtionality to it.