How to grant remote access to MySQL for a whole subnet?

You would just use '%' as your wildcard like this:

GRANT ALL ON *.* to root@'192.168.1.%' IDENTIFIED BY 'your-root-password';

It looks like you can also use a netmask, e.g.

GRANT ... TO 'user'@'192.168.0.0/255.255.255.0' IDENTIFIED BY ...

EDIT: Consider looking at and upvoting Malvineous's answer on this page. Netmasks are a much more elegant solution.

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Simply use a percent sign as a wildcard in the IP address.

From http://dev.mysql.com/doc/refman/5.1/en/grant.html

You can specify wildcards in the host name. For example, user_name@'%.example.com' applies to user_name for any host in the example.com domain, and user_name@'192.168.1.%' applies to user_name for any host in the 192.168.1 class C subnet.