You may always get information about capabilities in use opening “Admin menu” window for the “Administrator” role. Menu items under “Appearance” menu are protected with ‘edit_theme_options’.
Add ‘edit_theme_options’ to your role, update it. Then open “Admin menu” for this role and block unneeded menu items for it.
Thanks Vladimir for pointing me to the “edit_theme_opions”. Adding this to the role added the options I was looking for in the “Admin menu” area. Great help – thanks!