How can I access block field values in mytheme_preprocess_block?
Here is a code snippet that we use to create block template suggestions based on the block content type (bundle), doing something similar based on a field should be fairly easy (...->yourfield->value):

/**
 * Implements hook_theme_suggestions_HOOK_alter() for block templates.
 */
function ourtheme_theme_suggestions_block_alter(array &amp;$suggestions, array $variables) {
  if (isset($variables['elements']['content']['#block_content'])) {
    $suggestions[] = 'block__' . $variables['elements']['content']['#block_content']-&gt;bundle();
    $suggestions[] = 'block__' . $variables['elements']['content']['#block_content']-&gt;bundle() . '__' . $variables['elements']['content']['#view_mode'];
  }
}