Try something like this:

print "Stopping network...\n";
system('/etc/init.d/network stop 1>/dev/null 2>&1');
print "Starting network...\n";
system('/etc/init.d/network start 1>/dev/null 2>&1');
[download]

2>&1 redirects STDOUT to the same place that STDERR is going, ie. to /dev/null.

Update: However, you probably should think about doing something useful with STDOUT and STDERR, such as redirecting the output to a logfile. That way, you have somewhere to check if something goes wrong.

Hope this helps,
Darren :)