Table-based layout was already falling out of favour before this happened. Even when you had fixed sites that were predominantly desktop, divs and spans (coupled with CSS) made life a lot easier to put things where you wanted them and lay out a site faster. It was also cleaner when reading the markup.
From what I remember, the move away from tables happened around the same time external CSS took off and you could partition your site with reusable styling instead of framesets or tables.
Responsive design then sped up the process a few years later.