well, you can write it a little bit shorter...

my $count = 0;
$_ and $count++ for ($x,$y,$z);
if (1 == $count) {
  print "Exactly one variable set\n";
}
[download]

or or even pack the $count and ++ into a do:

if (1 == do {my $cnt=0; $_ and $cnt++ for ($x,$y,$z); $cnt } {
  print "Exactly one variable set\n";
}
[download]

But I doubt that this is more readable

update: I'd prefer a grep solution as posted by tye

if (1 == grep {$_} ($x,$y,$z)) {
  print "Exactly one variable set\n";
}
[download]