I have not used tables to lay out the non-tabular content of a website for many years, so I might have missed a few things here, but I have some alternatives and ideas.
Distract him from some: it seems that the main problem is that your boss wants you to use a web development technique that is faster than the one you are currently using, allows you to achieve the same layout, and it is not related to semantic markup.
I think that to achieve the same goals, you can use CSS or a platform for building sites, such as Twitter Bootstrap or 960gs instead of a tabular layout (Note: 960gs is included in Bootstrap). These structures have some non-semantic markup, such as a div, to contain lines and spaces for setting width and offset elements, but better than using a table in terms of availability and amount of non-semantic markup.
You can further mitigate this by specifying your element identifiers and additional classes and styling them, and there is less non-semantic markup than if you were using a table-based layout.
Based on my interpretation of the root problem, a structure like any of these also gives you elements with a pre-styled style and a way to nicely highlight elements that will save your time in the overall design -> code -> loop revision and none of this contradicts web best practices -development.
Some resources for Twitter Bootstrap:
http://twitter.github.com/bootstrap/ - Download and good documentation
http://twitter.github.com/bootstrap/scaffolding.html - examples of code that you would use in Bootstrap instead of a table layout
960gs (960px wide mesh):
960.gs/- Homepage
https://speakerdeck.com/u/nathansmith/p/960-grid-system - the final tutorial on 960gs and the reasons for using it
http://sixrevisions.com/web_design/the-960-grid-system-made-easy/ - A tutorial that I first used to learn about mesh systems in web design
If my initial guess is wrong, sorry! Also, if you have any questions or want more information, let me know.
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