If you have Ruby, you can install the mysql2xxxx gem (not the mysql2json gem, which is a different gem):
$ gem install mysql2xxxx
and then run the command
$ mysql2json --user=root --password=password --database=database_name --execute "select * from mytable" >mytable.json
The gem also provides mysql2csv and mysql2xml. It’s not as fast as mysqldump, but also doesn’t suffer from some of mysqldump’s weirdnesses (like only being able to dump CSV from the same computer as the MySQL server itself)