Angmar at the end of the Third Age
 

According to the appendices, Angmar was destroyed by the allied forces of Gondor and the Elves. After that we hear no more about it. What do you think happened to the people of Angmar(Orcs, Hillmen of Rhudaur) afterwards? Knowing the way of evil in Middle earth, I would venture to say they multiplied in number after the War and reoccupied Carn Dum. As such, I find it strange that Angmar never figured much in the War of the Ring. Any ideas?

While its not explicitly said, I think that the people of Angmar mostly died out. While Tolkien never takes us through the specific land of Angmar he paints a clear picture of desolation through that region.

I think the failure to mention any evil stirring in Angmar during the War of the Ring is also an indication that the area was deserted.

We know that Sauron's plan after he started to gather power again was probably to reoccupy Angmar and thus threaten Eriador from the North, using the Easterlings (the battles of Erebor by the time of the War of the Ring) and if he could, then Smaug (on more about this topic see Unfinished Tales, The Quest of Erebor). These plans never came true, while the explanations may be many.
One thing we know is that the Orcs were quite numerous in the Misty Mountains, especially Gundabad, which is not that far from Angmar. But personally, I would imagine it the way that all the remaining Hillmen or men of Angmar were unfriendly to the Orcs, they served their own chieftains and as long as no one united all of them under one banner (which did not happen at that time), they posed no threat to the West. I imagine it the way that the goblins of Mount Gram from time to time made raids on the Hillmen, as much as the Goblins on the east side of the Mountains did to the Woodmen, and there was probably little friendship between them. From the south, the Ettenmoors were a dangerous and wild place for everyone, and the "raiders" were scarce and unorganised (cf. Bert, Tom and William). There was also nothing to raid, except for Rivendell (which would be a suicide unless the army was really big and organised) or Bree (which was too far from Angmar). The most daring movement was spotted during the Fell Winter, when the goblins of Mount Gram entered Northfarthing and were beaten back by Bullroarer. Also, when it was up to their skill, the Rangers (and some Elves) were patroling the wilderness, which would probably serve its purpose in spotting and eliminating medium-sized raiding parties that could have formed.
Concerning Carn Dum itself, we don't know whether anyone lived there after the fall of Angmar. Given the nature of the subject, it is possible that the place was abandoned and it became a forsaken place, in the minds of the Hillmen and Orcs alike surrounded by tales of shadows and evil spirits dwelling there... (and these tales could have been even true).
Whatever the case, there was no connection between Mordor and Angmar anymore after the fall of Witch-King's realm.

That's a bit of concluding from what we know. Personally, I am in progress of making an RPG which partially takes place in Angmar around the year 2912 TA and I have made a lots of stuff concerning that, but that are only my own fantasies and are in no way reliable :)

I thought Angmar became ruined and the remnant of its people scattered across Middle-Earth, until they sort of merged with the other folk of Middle-earth. That or maybe the Witch-King had the people of Angmar move to Minas Morgol, Dol Guldor, or Mordor and they became a part of sauron's army.

I'm sure it is mentioned somewhere that the Eotheod destroy all the men and orcs of that realm east of the mountains.

Yes. I remember reading that as well. Gondor and its northern allies destroyed the forces of Angmar on the west side of the mountains.

But that makes me wonder...what happened to children of the human Angmariansafter their fathers(/parents?) were destroyed by the Eotheod and Gondorians?

It will be quite logical that only men fought in the army of Angmar, the women most probably remained at home with children... for the Hillmen of Rhudaur, the few survivors probably returned home to the few settlements in the woods, or maybe (and that would probably go for those who lived in Carn Dum in particular) the survivors (men, women, children) migrated somewhere else - to the South, to Bree*, from Angmar it also isn't far to the Lossoth who could have accepted the women and children. Maybe some good-hearted Dúnedain could adopt children of the Hillmen who lost their parents?* But otherwise, many of the survivors - even if they were migrating - would probably also die in the wilderness. I think the tale itself will be quite sad.

*Wow... I just got a fantastic idea for a tale or an RPG. What if a child of some chieftain of the Angmarim was fostered and growing up in Rivendell, as it happened with some Dúnedain? The life-story could be definitely interesting...

Shouldn't this be in the Role Playing Game section?

"Cirion and Eorl" (Unfinished Tales) it is mentioned in note 19 that the Eotheod "drove away the remnants of the people of Angmar on the east side of the Mountains."