In a comment on a response to this question, the following result is vaguely stated: Suppose $X/\mathbb{Q}$ is a surface which fibers into genus 2 curves over a base curve $C/\mathbb{Q}$; then computing $H^2_{et}(X)$ is equivalent to computing $H^1_{et}$ of a 125-fold cover of $C$. Would someone be willing to explain this further? Given the importance of etale cohomology and the difficulties in working with it, I would really love to see the details of this spelled out.