We know Sauron bent most the orcs of the 3rd age to his will, but were the men of Khand, Rhun and Harad bent to his will as well or simply bribed?
This depends on what you mean by "bent to his will".
We read in Of the Rings of Power and the Third Age:
In the east and south well nigh all Men were under his dominion, and they grew strong in those days and built many towns and walls of stone, and they were numerous and fierce in war and aimed with iron. To them Sauron was both king and god; and they feared him exceedingly, for he surrounded his abode with fire.
This is the clearest statement that I am aware of regarding Sauron's domination of these peoples in post-LotR writings, but - of course - "under his dominion" could have many meanings - magical, political, coercion, bribery (presumably with Rings), etc are all valid interpretations.
If we go back to the Lay of Leithian (History of Middle-earth 3) we read:
Men called him Thû, and as a god
in after days beneath his rod
bewildered bowed to him, and made
his ghastly temples in the shade.
Not yet by men enthralled adored,
now was he Morgoth's mightiest lord...
This is a pre-LotR writing (easily identified by it's use of "Thû" rather than "Sauron") so the geographical expansion and tales of the Second and Third Ages didn't yet exist when it was written, but it's use of words such as "bewildered" and "enthralled" certainly suggests some form of magical domination.