Getting Links in Multiple Categories (intersection)

It should be as simple as that: Just get the terms from the link_category and use the get_terms() argument instead, which should work like you need it.

get_terms(
     'link_category'
    ,array(
         'include'      => array( 1, 2, 3 )
        ,'orderby'      => 'name'
        ,'order'        => 'ASC'
        ,'hierarchical' => 0
     )
);