The Rule of Thirds in Design

I am in the process of trying to redesign a visualization.  The layout is bugging me, but I'm not sure how to deal with it.  That got me to take a few moments to crack open google on graphic design.  Are there magic rules that probably can help no matter what?

Seems so.

A key factor at play is how the eye naturally tends to read a web page, or a photo, or a painting... or actually every single scene of life we visually experience(!). The Rule of Thirds is a principle that describes how this manifests itself: our eyes tend to go to four specific locations when they scan a scene, these locations being the inner intersection points of a 3x3 grid overlaying the scene.

Carrie Cousins has several blog posts on this (Using the Rule of Thirds as a Grid Outline, Understanding the Rule of Thirds in Web Design) that particularly resonated with me.  Here is one of her images I found particularly useful:

Where Our Eyes Focus on a Scene (from Carrie Cousins, Using the Rule of Thirds as a Grid Outline) 

It would seem fairly ineffective to put together any visualization without keeping these things in mind, simply because it is important to know where a user's eyes will focus... whether or not we want to put anything interesting there.  To quote Cousins,

"You can work with or against the rule of thirds, but you can’t eliminate it."