I'm currently reading Zen Mind Beginners Mind which I am enjoying. However when I read Zen books I'm always confused about the role of effort in zazen practice. Reading the book it makes it seem like there is no effort involved. In one sense I can subscribe to this. However if zazen was truly effortless then everyone would be doing zazen most of the time. That seems like a flawed conclusion to me. Can anyone help me with how effort relates to zazen please.
As someone who has practiced zazen (in the Rinzai tradition) for many years, I will second Michael's answer. That very much describes how I meditate. You have to lift and apply the mind in meditation otherwise, well, otherwise you tend to fall asleep. ;-) Too much effort (like forcefully eliminating thoughts and 'crushing' the mind on the meditation object) is also a bad idea. The approach might be slightly different for the Soto lineage (and the language your book uses seems to suggest a Soto bent). Still, I highly doubt there's that much difference in how the mind is engaged while sitting.
One thing to remember is that Zen is heavily influenced by Taoism. You might want to tangentially exploring the concept of non-effort by reading up on the Taoist concept of wu wei. Sometimes it helps to explore a similar idea from a different avenue.
I am not familiar with zazen since I practice theravada meditation. In theravada there is the idea that you have to make effort to sit and meditate. You also have to make effort to establish the 3 mental qualities which will support the meditation. Those are mindfulness, clear comprehension and ardency (sati, sampajañña and atapi in pali). But having established those qualities the meditator has to let go of all the rest and that translates as non-doing or in other words not effort. If one continues to apply effort at this stage the meditation becomes very tiring with little progress. Hope that helps.