From the Teheran Times - seems a subject (unlike some) on which we can accept Iranian press reports!!
"The Shushtar waterworks comprises bridges, dams, mills, qanats, reservoirs, tunnels, and canals, most of which were constructed in the Sassanid period (224-651 CE), especially during the reign of Shapur I (241-272 CE).
However, some structures of the system date back to the Achaemenid period!"
So UNESCO's "can be traced..." is strictly correct but it is primarily Sassanid as Wiki says.
Given the relatively common area occupied by the heartland of all the "Persian Empires" I suspect that some sites are going to have some remains (even if not very significant) from several dynasties but I believe your 3 are all primarily/significantly Achaemenid.
Herewith a map of the Achaemenid empire. http://www.bestirantravel.com/culture/history/empiremap.html
And here one of the Sassanidhttp://en.wikipedia.org/wiki/File:Sassanid-empire.gif
Here a dynasty list!http://www.bestirantravel.com/culture/history/dynasties.html
But is there a "Connection" for "Sassanid"? Needs 2 more after Shushtar.
Bisotun is claimed to have Sassanian remains.
For Persepolis - The Sassanian town of Istahkr was nearby but the most famous Sassanian carvings are at the T List site of Naghsh-é Rostam - again nearby. Wiki says of these Sassanid carvings "... as well as a monumental inscription in Persian and Greek in the vicinity of Persepolis." But I can't find out if this "vicinity" was actually inside the inscribed boundaries of Persepolis - it could refer to Istahkr. My memory of my own unguided visit of 38 years ago is of no help I am afraid!
For Passargardae - I know of no Sassanian connection - it was already overtaken as a city by Persepolis in Achaemenid times.