From http://stackoverflow.com/questions/9468059/magento-filter-product-collection-by-multiple-categories

new_gist_file_0

$category = Mage::getModel('catalog/category')->load(100);
$allChildsIds = $category->getAllChildren($category);

$visibility = Mage::getModel('catalog/product_visibility');

$collection = Mage::getResourceModel('catalog/product_collection');
$collection = $this->_addProductAttributesAndPrices($collection)
  ->addStoreFilter()
  ->setFlag('do_not_use_category_id', true)
  ->setFlag('disable_root_category_filter', true)
  ->addAttributeToSort('created_at', 'desc');

$whereCategoryCondition = $collection->getConnection()
  ->quoteInto('cat_index.category_id IN(?) ', $allChildsIds);
$collection->getSelect()->where($whereCategoryCondition);

$conditions = array();
$conditions[] = "cat_index.product_id = e.entity_id";
$conditions[] = $collection->getConnection()
  ->quoteInto('cat_index.store_id = ? ', Mage::app()->getStore()->getStoreId());
$conditions[] = $collection->getConnection()
  ->quoteInto('cat_index.visibility IN(?) ', $visibility->getVisibleInCatalogIds());

$collection->getSelect()->join(
  array('cat_index' => $collection->getTable('catalog/category_product_index')),
  join(' AND ', $conditions),
  array()
);

$collection
  ->setPageSize(3)
  ->setCurPage(1);

$collection->load();