The key is the function: load_theme_textdomain. This function is placed last in functions.php of the Theme.
The .mo files must use language-only filenames, like languages/de_DE.mo in your theme directory.
Unlike plugin language files, a name like my_theme-de_DE.mo will NOT work. Although plugin language files allow you to specify the text-domain in the filename, this will NOT work with themes. Language files for themes should include the language shortcut ONLY.
I have changed a little in functions.php of the theme, so load_theme_textdomain now says:
load_theme_textdomain ('veryplaintxt' get_template_directory().'/lang');
Because then I can place my language file in the folder: themes/veryplaintxt/lang - I like to keep order