Aren't we all missing the obvious answer given to us by the compiler?
You just need to use no strict 'refs';.

...
{
 no stict 'refs';
 for (1..3) {
     my $ref = \${'str'.$_};
     print "The string is $$ref"; 
 }
}
...
[download]

or, as other people suggested, just use a hash.