Term::ReadKey is the only solution that I could get to work. It is unfortunate that it needs a non-standard module. Here's my code:

use Term::ReadKey;

print 'Enter Username: ';
my $user = <STDIN>;
chomp $user;

print "Enter Password: ";
ReadMode ('noecho');
my $pass = <STDIN>;
chomp $pass;
ReadMode ('restore');
print "\n";
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Update: link to kobesearch since search.cpan is down