For those following along, here is the solution that worked, based on Randy's advice:

use ExtUtils::MakeMaker;
# See lib/ExtUtils/MakeMaker.pm for details of how to influence
# the contents of the Makefile that is written.
my @list_of_db_files = ('db\ECO','db\NIC','db\Opening');

WriteMakefile(
    'NAME'        => 'Chess::PGN::EPD',
    'VERSION_FROM'    => 'EPD.pm', # finds $VERSION
    'PREREQ_PM'        => {}, # e.g., Module::Name => 1.1
    ($] >= 5.005 ?    ## Add these new keywords supported since 5.005
      (ABSTRACT_FROM => 'EPD.pm', # retrieve abstract from module
       AUTHOR     => 'Hugh S. Myers <hsmyers@sdragons.com>') : ()),
);

sub MY::post_constants {
    join("\n",
        'INSTALL_DB_FILES=$(INSTALLSITELIB)\Chess\PGN\db',
        "DB_FILES=@list_of_db_files");
}

sub MY::postamble {
    my $self = shift;
    my @text;

    push(@text,
        'install :: install.dbfiles',
        '',
        'install.dbfiles:: $(DB_FILES)',
        "\t\$(MKPATH) \$(INSTALL_DB_FILES)",
        "\t\$(CP) \$(DB_FILES) \$(INSTALL_DB_FILES)");
    join("\n",@text);
}
[download]

–hsm

"Never try to teach a pig to sing…it wastes your time and it annoys the pig."