[Inziladun]

The White Council looks to be a gathering of those who saw the 'big picture' in Middle-earth; those with the most perspective and long experience. The Dwarves were always guilty (save in extreme necessity such as the Nirnaeth Arnoediad against Morgoth, or the Last Alliance) of being very self-involved. I can't picture the Dwarves of Erebor taking the time away from their own affairs to recognise the importance of council with other powers in Middle-earth. It was only immediately before the start of the War of the Ring that they were forced to do so.
The same could be said of the Dúnedain, at least those of Gondor in the Third Age. Boromir, for instance, thought only Gondor stood as a barrier to evil in the West, and its only allies were the Rohirrim. Gondor had come to the point where they believed they had no other allies, and it was only Boromir's dream and Denethor's anxiety about Mordor that brought a high Man of Gondor to the Council of Elrond.
As for the Dúnedain in Arnor, they would have been too busy keeping guard, and Aragorn probably wouldn't have thought himself deserving of a seat on the Council anyway.
The Council had to consist of those who could govern the 'real' war against Sauron without thought of any individual conflicts or interests. The greatest remaining Eldar and the Istari seem to be the closest to that ideal.