LinqPad is basically a gui front end for MS-SQL Server. In that sense, you have several options, but remember that you'll need to install MySQL server (
sudo apt-get install mysql-server in terminal) to use them. Additionally, you'll probably want create and populate a database to play around with. The biggest difference will be that you won't be able to use Linq syntax, because Linq is a .Net syntax (you can, however, get similar syntaxes by getting into some of the languages and frameworks, the most closely-resembling probably being Ruby or Python).
MySQL's built-in terminal application. You'll have to install mysql-client (which you'll probably need if you do any PHP/Python/Ruby development, anyway), but then, you can open up a terminal and type in
mysql -u[a mysql user] -p (
-p if the mysql user has a password, otherwise just leave the
-p off). That will bring up a mysql console, which you can then run SQL queries directly in.
MySQL Workbench This is a more full-featured database management GUI more similar to Management Tools than LinqPad, but it features a SQL query editor that allows you to run queries to your heart's content. You can find this in the repos as the
PHPMyAdmin This is somewhere between, and is a web-based solution. You'll need to install the
php5-mysql packages, and a web server (such as lighttp or apache, either of which PhpMyAdmin should give you the option to configure), then install the
phpmyadmin package. Open your browser and navigate to the URL you created for it, and you can play with SQL in the "SQL" tab, again to your heart's content.
I personally prefer Workbench, or just the plain terminal application for most things.