Build a new template that is not a table-based layout. Name your editable regions the same in the new template as they were named in the existing one. Apply the new template, one-by-one to each existing page.
Alternatively, after creating and saving your template as above, close ALL files. Then do a site-wide Find and Replace to find the old template's file name, and replace it with the new template's file name. Now open the new template, make a trivial change to it, and save it. Allow all files to be updated.
Please make a backup copy of your entire site before attempting either of these two.
