5th Dunlending Armoured Division, Front and Center!
 

Following the victory of the Rohirrim at Battle at Helm's Deep in the Third Age, King Théoden magnanimously allows the Dunlendings to leave in peace after they help clean the place up a bit.

Now, would this act have changed any of the hearts of these men, making them want to join up against the Enemies of Men?

These were presumably tough fighters, even if ill equipped, as they had to train and fight beside the Uruks of Saruman. Maybe (or not) none rode off to the battle down at Mundburg, but is it possible that they helped later against the invasion of Rohan?

Would Sauron cared that these men had once been in the sway of Saruman? And would the Dunlendings have realized that they needed to stand up against the Darkness?

My assumption has been that the Dunlendings learned Saruman had largely lied about the Rohirrim, but he had also enlarged upon historical grudges the Dunlendings had against them. Just finding out the Rohirrim didn't execute their prisoners probably wouldn't have been enough to induce them to fight for their long time enemies. I would think that as soon as they had finished the work the Rohirrim set for them, they had simply gone back home and taken no further part in the War.
As to whether, in time, the Dunlendings would have allied themselves with the West in the Fourth Age, that's possible, but I think it probably would have taken a policy of active efforts of reconcilliation on the parts of Éomer and his successors.

Somewhere in HoME in an early draft of LoTR Theoden receives volunteers from amongst the Dunlendings if I remember. So possibly!

Whether Tolkien decided against this idea or simply didn't think it important enough to include I don't know. I'd imagine that they would not make up a major force, as although they had some cavalry there don't seem to be many, so could not contribute that much to the ride to Gondor.

Sure they could have helped garrison Rohan, but the Rohirrim would have to be very sure they would not re-revolt when their backs were turned! So probably safer to send them home.

After the war, maybe Aragorn would be best to bring the Dunlendings into the fold, as he is not a traditional enemy and is also not blonde!

The essay The Battles of the Fords of Isen in UT does indicate that after the return of the King the Dunlendings' hatred was 'appeased'. I would still think it unlikely any of them participated in the War of the Ring after the defeat of Saruman.

Note that these Men were akin to those that inhabited the Paths of the Dead. You'd think that that lesson would spur them just a little.

Given that Dunlendings had grievances against the Rohirrim
not unlike that of Native Americans and Aboriginals against
Western colonizers, actively helping Rohan without some sort
of at least token territorial "readjustment" seems unlikely.