I agree, while(1) loops can be bad. I'm fond of doing something similar to what you are doing instead of while(1). I have spent much more time than I care to remember debugging code that was due to poor while(1) implementations.

I have a few minor nits to pick with your example.

1. You don't close any of your file handles. Why keep em around once you are done? I would inject a close right before your delete
2. Intead of repeatedly checking every file in the list you are processing, why not just check the list of files currently open? The foreach my $file (@files) loop is easy to swap out with a foreach my $fh (keys %fh)

The change would look something like this:

while (keys %fh) {
   foreach my $fh (keys %fh) {
      if (defined(my $line = <$fh>)) {
         chomp $line;
         print $line;
      } else {
         close($fh);
         delete($fh{$fh});
      }
   }
}
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