After a frank exchange of ideas with my brother I made another attempt at using DIVs with CSS and the float attribute in CSS. I was unable to get Mozilla to render the content in the way I wanted it to display. Interestingly, IE did what I wanted.
With TABLE tags, the site still looks great in links and lynx, two text-based browsers. It works slightly less well in screen readers; the one I tested was Simply Web 2000, a free screen reader. In fact, the only reason why I took another attempt at making the site table-free was to make it work better with screen readers.
For what it’s worth, I will probably try again later. Right now, I’m leaving the site design as is.