find /path/to/dir -name name_of_file_to_match | xargs /bin/rm
I use often the above command to clean a directory from all the back up files created by emacs, so for example if I am developing a site at:
/var/www/a-site/ then at the end of the day or at the beginning of a new one I clean up the directory as follow:
find /var/www/a-site -name '*~' | xargs rm
Be careful with the above command, specially if you run it as root, you could get the wrong regex and delete things you don't mean to.