You want to loop over the array index, you need to enumerate from 0 to the array length minus 1, which is that "$#". The initialization (my $i = $j = $k = 0;) is not required because you already use the "my" in each loop.

A more C stylish approach (not necessarily better):

#!/usr/bin/env perl 
use strict;
use warnings;

my @array1 = (1, 2, 3); #veryfast
my @array2 = (qw/a b c/); # Enough to test #veryslow
my @array3 = (qw/I II III/); 


# my $i = $j = $k = 0;
for(my $i=0;$i<=$#array2;$i++) {
  for(my $j=0;$j<=$#array3;$j++) {
    for(my $k=0;$k<=$#array1;$k++) {
      print  "$array1[$k],$array2[$i],$array3[$j],\n";
    }
  }
}
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Note that because in each iteration it needs to calculate $# these loops are slower than iterating over the values of each array, as already shown by other monks.