I've heard this from many different places. I think it makes no sense, but I was just wondering.
It's true, and has been since the invention of semi automatics. But its only true when you lump everything together in groups. Individual rifles of either type can be more accurate than other individual rifles of either type.
Do note the huge percentage of "accurate" semi autos mentioned in the responses are ARs. When talking about accuracy and semi autos, about the only one discussed these days is the AR (and variants). And its true, the current generation of the AR design is accurate, generally, it is the most accurate semi auto ever produced, to date.
Now add in all the other semi autos that aren't generally as accurate as the AR, (such as AKs, and dozens of other military and sporting designs) and I think your average accuracy for "semi automatics" will be a bit lower than taking all the bolt actions, military, sporting, & target, and doing the same.
Simply put, there are two basic things that impact the accuracy of semis vs. bolt actions.
First is the fact that accuracy is repeatability. The gun fires every shot exactly the same way. The harmonics of the rifle vibration during firing being as close to exactly the same every time, is what makes a rifle accurate, or not.
The greater the number of parts, and
moving parts, the more difficult this is to achieve. The more complex mechanism of a semi auto makes it more difficult to get uniformity than the bolt action. (More difficult is not impossible)
The other thing is simply, the overwhelming majority of semi autos were never designed with ultimate accuracy in mind. They were designed and built to be accurate enough to do the intended job. If a 3MOA rifle reliably puts meat in the freezer or takes down enemy troops, there's very little reason to spend the time & money to build a sub MOA one, IF it can even be done.