I don't think it's so much the use of tables, more the assumption that everyone has a wide screen.
If the gallery had the pictures with the top four in a two by two pattern, then the bottom three one beneath another, they'd still look gorgeous, but not need just over 3x the width of the bottom three to fit on more screens without side scrolling.
Similarly, the menu graphics (the 'About me', 'Donations'
in the fancy script) could be made smaller, take up less horizontal space, and still be readable.
For the other pages, changing the width of the tables from a fixed number of pixels to '80%' or '90%' will make it fill most of whatever width is available automatically.