What makes a smokeless jet engine? I was recently reading about the MiG-29 and its many upgrades, one of which (MiG-29M) was to smokeless engines.

What makes a smokeless jet engine? Is it just the fuel, or something about the way the engine works itself?

I've observed smokey engines were more common in the past and I'm wondering why that is too. Were they just easier to build for some reason?
Generally, smoke comes from incomplete burned fuel.

Normally, a jet engine burns a very lean mixture (due to turbine temperature constraints). Nevertheless, getting all the fuel to burn is a considerable challenge, primarily because the air/mixture flows faster in the engine than the flame front. So in order to provide stable burn, many tricks come into play: the combustion chamber expands after the compressor; there are special vortex generators that force the mixture to backflow and circulate; the air may be fed in stages along the chamber, etc.

This complicated gas dynamics is difficult to optimise, especially in the 70s when MiG-29's engines (RD-33) were designed. What usually happens is that there are pockets of over-rich mixture in the combustion chamber, and the extra fuel decomposes due to high temperature before it has an opportunity to burn.

From what I know, RD-33MK got a redesigned combustion chamber (amongst other things), but what exactly the changes are is hard to say. I can speculate that the source of the original problem was that RD-33 had an early example of the annular combustion chamber, which was not well studied then. But the problem was bad enough to make MiG-29 more easily detectable in visual fight.