This is what I eventually ended up with:

my @s = (1..168);
my @q;

{ $a=rand(@s);push(@q, $s[$a]); splice(@s, $a, 1); @s && redo }
print join(" ",  splice(@q, 0, 12)) . "\n" for (1..14);
[download]

kudos to mikfire for reminding me about the return value of splice.

Update: my splice was off by 1. whoops