The MySQL dump method would be the best bet if you're happy to re-import the whole database. For anyone that doesn't want to do this - WordPress core installation only actually consists of 11 tables, of which few are content columns, so doing a replace by column would be equally easy. Assuming you don't have loads of plugin tables referencing your link or string this would be your SQL:
UPDATE wp_commentmeta SET meta_value = REPLACE(meta_value,'xcurrentx','xreplacementx');
UPDATE wp_comments SET comment_content = REPLACE(comment_content,'xcurrentx','xreplacementx');
UPDATE wp_links SET link_description = REPLACE(link_description,'xcurrentx','xreplacementx');
UPDATE wp_options SET option_value = REPLACE(option_value,'xcurrentx','xreplacementx');
UPDATE wp_postmeta SET meta_value = REPLACE(meta_value,'xcurrentx','xreplacementx');
UPDATE wp_posts SET post_content = REPLACE(post_content,'xcurrentx','xreplacementx');
UPDATE wp_posts SET post_title = REPLACE(post_title,'xcurrentx','xreplacementx');
UPDATE wp_posts SET post_excerpt = REPLACE(post_excerpt,'xcurrentx','xreplacementx');
UPDATE wp_term_taxonomy SET description = REPLACE(description,'xcurrentx','xreplacementx');
UPDATE wp_usermeta SET meta_value = REPLACE(meta_value,'xcurrentx','xreplacementx');