I use a board (unless I'm flicking through the book purely for visualisation practice). Sometimes I will visualise variations if possible, just for convenience, but if the variation is too long to visualise comfortably then I will play it out on the board.
Some people are more gifted in visualisation than others, but I think that anyone can improve their visualisation with practice. Doing tactical puzzles will help here (for example, on Chess Tempo).
Very good post. I would also like to know the answer.