How does a woodpecker not give itself concussions?
Various features of brain,skull and beak anatomy help to achieve protection.
A paper was published in PLoSOne in 2011 on this very topic:
Why do woodpeckers resist head impact injury: a biomechanical investigation
There is also a very readable summary on the BBC website. I advise that you read the whole article, but here is a quotation which lists the main findings:

The team's simulations showed that three factors were at work in sparing the birds injury.
Firstly, the hyoid bone's looping structure around the whole skull was found to act as a &quot;safety belt&quot;, especially after the initial impact.
The team also found that the upper and lower halves of the birds' beaks were uneven, and as force was transmitted from the tip of the beak into the bone, this asymmetry lowered the load that made it as far as the brain.
Lastly, plate-like bones with a &quot;spongy&quot; structure at different points in the skull helped distribute the incoming force, thereby protecting the brain.
The team stresses that it is the combination of the three, rather than any one feature, that keeps woodpeckers pecking without injury.