I'm glad to see this topic being covered. I've maintained for a very long time that there is no reason a semi-auto *can't* be as accurate as a bolt action rifle. I'd like to share some of my thoughts, and see if any of them resonate.
A big part of it is technique, but some guns are less tolerant than others. The HK 91 comes to mind as one that requires much more vigilance and precision in technique. AR's are easier to shoot well, in my opinion, in part because they are more ergonomic. Also, their recoil impulse goes straight back into your shoulder. If the bore line is above your shoulder, things get more complicated.
Personally, I'd say that normally you should be leaning into the rifle a bit. That is, acting like a sort of spring-board, more-so than a lump of clay. Your position needs to allow you to "reset" automatically. That is, if someone were to tap straight back on the muzzle of your rifle with a rubber mallet, as soon as that mallet looses contact the muzzle is moving forward and stopping at (near) exactly the same point. The recoil of a full sized rifle cartridge is always going to cause you to move. It should always cause you to move in exactly the same way and to the same place.
I'm not going to tackle trigger control right now.
I am going to mention breath control though. A lot of people talk about making use of 'the respiratory pause.' I have never been able to make it work for me though, and I think there are others as well with the same problem. My technique ties in sight picture and trigger control like this:
My basic premise is that it's impossible to eliminate all movement, so it's better to make use of it. Dad told me once regarding off-hand, that if you could keep the sights making a side-ways figure-eight and start your trigger manipulation when the sights were furthest out, the shot would break when they were back in the center. I rejected figure eights, but ran with the constant movement.
I let my sight picture, whether iron or optical, move vertically over the target as I breathe. In particular, my sight picture is at its lowest -the bottom of the target - when I've finished my in-breath. At that point, I take up the first stage on my trigger. Then, as I'm breathing out the sight picture climbs the target as I steadily apply more pressure to the trigger - without trying to make the sights stop at the center of the target. It doesn't take long to figure out the timing, to make the shot break when the sights are over the bull. As a side note, my martial arts training emphasized projecting force in time with exhalation. So, it helps me both in driving the muzzle toward the target and the mental aspect of sending lead down range.
That technique results in a lot of X's, with bolt guns and semi-autos ...for me. There are a lot of people it just doesn't work for. It's also not the same with a single stage trigger, or a lightweight bench rest type trigger.
That's all for now, but I do have more if you're interested. I'd love help.
PS: I have some thoughts on mechanical accuracy improvements in semi-autos of most types as well. If those would be more appropriate elsewhere, that's not a problem at all.