Your grep solution would be intuitive to many monks, but it becomes more maintainable when paired with well-named variables or subroutines. My (current, and PBP-influenced) preferences:

• If you check only once in your program, then be descriptive with a variable.

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

• If you check more than once, sometimes comparing the count to values other than one, then use a sub named for the count it returns.

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

• If you check more than once, always comparing to one, then use a sub named for that comparison.

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