First of all, this is a good thing. "Quirk mode" and other lame must die. But you should support all major browsers, not just IE. As I just said, remove all incompatibilities and do everything according to W3 standards, but you design should only depend on the features surely implemented on all major browsers.
Now, the thing is: these days you cannot cover all possible incompatibilities between browsers and compensate for them. You just cannot win the race. So, what to do? My answer is: focus on main goals of your design and make the rendering not too sensitive to small rendering detail. How to do it? First of all, by fine setting of your goals and requirements to the site.
I tried to express basic ideas in my past answer:
webpages layout change when running in different browsers.[
^].
Good luck,
—SA