You need to set a Content-Type header, and use the special CGI.pm parameter name POSTDATA to access the body of the POST request. This works for me:

Server:

#!/usr/bin/perl -T
use warnings;
use strict;
use CGI;
use CGI::Carp qw(fatalsToBrowser);
use Data::Dumper qw(Dumper);
use JSON::MaybeXS qw/decode_json/;

my $q = CGI->new();
print $q->header;
my $data = $q->param('POSTDATA');
$data = decode_json($data) if $data;
print Dumper($data);
[download]

Client:

#!/usr/bin/perl
use warnings;
use strict;
use LWP;
use HTTP::Request::Common;
use JSON::MaybeXS qw/encode_json/;

my @AoH = (
    { husband => "barney", wife => "betty", son => "bamm bamm" },
    { husband => "george", wife => "jane",  son => "elroy" },
    { husband => "homer",  wife => "marge", son => "bart" } );

my $data = { data => \@AoH };

my $url = 'http://myurl.com/cgi-bin/memo.pl';
my $ua  = LWP::UserAgent->new;
my $request = POST($url, Content_Type=>'application/json',
    Content => encode_json($data) );
my $response = $ua->request($request);
print $response->content, "\n";
[download]

Output on client:

$VAR1 = {
          'data' => [
                      {
                        'son' => 'bamm bamm',
                        'husband' => 'barney',
                        'wife' => 'betty'
                      },
                      {
                        'son' => 'elroy',
                        'wife' => 'jane',
                        'husband' => 'george'
                      },
                      {
                        'husband' => 'homer',
                        'wife' => 'marge',
                        'son' => 'bart'
                      }
                    ]
        };
[download]

Perfect. Learnt a lot again here! Thank you.