What actions did third world nations that wished to remain unaligned take?

They formed the Non-Aligned Movement.   
A conference of leaders of postcolonial nations of Asia and Africa met in Bandung, Indonesia in 1955.  This is often seen as the commencement of the Non-Aligned Movement (NAM).  One of the key leaders of the movement was Jawaharlal Nehru of India.  In a speech at the Bandung Conference, Nehru said, "“If all the world were to be divided up between these two big blocs, what would be the result? The inevitable result would be war. Therefore every step that takes place in reducing that area in the world that may be called the unaligned area is a dangerous step.”