I'm a bit curious as to what "cross-browser compatibility problems" you'd see with the code above. I ran a browsershots.org scan on that code, you can see the results. Of their default scan of 52 browsers, only an early pre-release version of Dillo does not display exactly as expected.
There are spacing issues which occur. Some browsers have extra space between the border of the frame and the div tag. Sometimes it runs over the size of the screen, which makes the need for a scroll bar on your window. It works in either IE or Firefox ( I forget which), but not the other.
This is why people go to great lengths to find a solution for this problem. I wish it was this simple. I've tried this approach before, and tried modifying it many times to meet my needs before giving it up and trying a less conventional solution (like adding backgrounds to the HTML tag and BODY tag).