don't worry = ไม่ต้องกังวล

do not = yàa / อย่า worry = kangwon / กังวล don't worry = yàa kangwon / อย่ากังวล

It depends on the context. Here are some samples: 1) "Don't worry, I'm going to pay the bill." In this case, "don't worry" would be best translated as "ไม่เป็นไร". This is the best translation of "don't worry" in most contexts 2) "Don't worry, he isn't going to die." In this case, "อย่าคิดมาก" or "อย่าคิดมากเกินไป" would be best. The literal translations you were given by others can also be used, but they aren't nearly as common in normal speech.

right, ไม่ต้องห่วง sounds more natural for me to be spoken in general. ไม่ต้องกังวล sounds more literary and official.

ไม่ต้องห่วง