I've tracked down what was causing this problem on my CZ-82, and since I haven't seen a post with all the relevant information in one place I'm going to write one.
Starting off, here's how the firing mechanism works (numbers refer to the numbering in the instruction manual):
Pulling the trigger (14) pushes the trigger bar (17) back. As it goes back, it in turn pushes on three things: the sear (29), the automatic (hammer block) safety (21), and the hammer lever (28). The automatic safety has no further relevance, so I won't be mentioning it again.
Whether the hammer lever or the sear is relevant depends on whether you are in double action or single action. In double action, the trigger bar pushes the hammer lever, which in turn pushes the hammer (26) back. As the trigger bar goes back, it is forced downward by the trigger bar disconnector (24). At some point it slips off the hammer lever, which releases the hammer and the pistol fires.
In single action, the hammer is already cocked and the hammer lever is unimportant. The trigger bar pushes against the sear, which release the hammer to fire.
The "drops into double action" problem arises when the trigger bar hasn't been forced downward far enough by the trigger bar disconnector by the time the sear is released. In this case, the hammer only goes forward a little bit, and the hammer lever is caught by the trigger bar. When this happens, the pistol reverts to double action.
This can be caused by either a worn trigger bar or a worn trigger bar disconnector (and I wouldn't be at all surprised to learn that various pins could wear and put enough slop in the trigger bar that they could cause it as well). Most of the people reporting the problem seem to have eventually found a worn trigger bar. In my case, it was a worn trigger bar disconnector.
The difference between the five trigger bar disconnectors available is in how quickly they force the trigger bar down (and hence where the gun fires). A lower number means it fires earlier. I ordered one of each trigger bar disconnector and two rivets (25), and kept trying different trigger bar disconnectors until I found one that would fire single action reliably (this is obviously a real YMMV, but in my case it was #3). You'll want to be very careful keeping them separate; there are no marks distinguishing them and the difference in their contours is so slight as to be invisible to the eye -- it's only a couple of thousanths of an inch.
To try the various trigger bar disconnectors, I just put the rivet through the frame and put the trigger bar disconnector over it and held it with my fingers while I worked the trigger (note that it's a lot easier to position the trigger bar disconnector when the safety has been removed). The trigger bar disconnector and rivet are a very tight fit -- it's almost, but not quite, an interference fit. Once I thought I had the right one I taped it down with painter tape and tried the trigger a few more times; then I peened the rivet, tried it a few more times, reassembled the pistol and took it to the range.
Anyway, I hope this saves somebody else the money I spent on parts that seemed a bit worn but had nothing to do with the problem!