That's an interesting approach. I still doubt it will complete in the life of the universe though.

This is my random generator/validator for approximating the percentage of valid patterns. It is 3.5 times faster than your original, but still if all the clouds in all the world, (a veritable storm), ran this and nothing else it would still take longer than the life of the universe to date.

#! perl -slw
use strict;
#use Math::Random::MT qw[ rand ];

$|++;

our $I //= 1e6;

my( $tried, $good );

for ( 1 .. $I ) {
    ++$tried;
    my @b = map int( rand 6 ), 1 .. 64;
    $good += checkBoard( \@b );
#    displayBoard( \@b ); print "$good of $tried"; <STDIN>;
    printf "\r%10u %18.15f ", $tried, $good / $tried unless $tried % 1
+0000;
}


sub checkBoard {
    my $ref = shift;
    for(
        [0..7],[8..15],[16..23],[24..31],[32..39],[40..47],[48..55],[5
+6..63],
        [ 0, 8,16,24,32,40,48,56],
        [ 1, 9,17,25,33,41,49,57],
        [ 2,10,18,26,34,42,50,58],
        [ 3,11,19,27,35,43,51,59],
        [ 4,12,20,28,36,44,52,60],
        [ 5,13,21,29,37,45,53,61],
        [ 6,14,22,30,38,46,54,62],
        [ 7,15,23,31,39,47,55,63],
    ) {
        my $n = 0;
        join( '', @{$ref}[ @$_ ] ) =~ m[(.)\1\1] and return 0;
    }
    return 1
}
sub displayBoard {
    print for unpack '(A16)*', join ' ', @{ $_[ 0 ] }
}
[download]

I *think* I now know how to write a program to generate a calculation that will produce the count. Your original program will be useful for testing the calculation against (much) smaller test cases. If it gets close for them, and results in a percentage of the 6^64 possibilities of 8.972% (approximation from 100 million trials), then it will be reasonable to assume the calculation is um...reasonable :)