OK, here's what worked for me, in 5 steps:

Step 1: don't worry about /etc/odbc.ini -- you don't need to create or modify it to make your MS SQL connection work from cygwin.

Step 2: you also don't need to worry about /etc/freetds/freetds.conf -- the defaults work out of the box

Step 3: set up your /etc/odbcinist.ini file like so:

[FreeTDS]
Description     = TDS Conection
Driver          = /usr/lib/cygtdsodbc.dll
Setup           = /usr/lib/cygtdsodbc.dll
UsageCount      = 1
FileUsage       = 1
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Step 4: write your Perl code like so (change as desired) #TMTOWTDI

#!perl
use strict;
use warnings;
use 5.020;

use DBI;

my ( $user, $pass ) = ( 'username goes here', 'password goes here' );

my $dsn =
{
   driver   => 'FreeTDS',
   server   => 'my.server.host.name',
   database => 'name_of_db_here',
   port     => 1433,
   tds_ver  => '8.0'
};
$dsn = sprintf 'dbi:ODBC:DRIVER={%s};SERVER=%s;database=%s;port=%s;tds
+_version=%s;',
   $dsn->{driver},
   $dsn->{server},
   $dsn->{database},
   $dsn->{port},
   $dsn->{tds_ver};

my $dbi_opts =
{
   PrintError     => 0,
   RaiseError     => 1,
   AutoCommit     => 1,
   LongReadLen    => 24 * 1024,
   LongTruncOk    => 1,
   odbc_utf8_on   => 1,
};

my $dbh = DBI->connect ( $dsn, $user, $pass, $dbi_opts );

my $rows = $dbh->selectall_arrayref( 'SELECT * from some_table_you_kno
+w' );

use Data::Dumper;

say Dumper $_ for @$rows
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Step 5: Profit!

It turns out that this wasn't nearly as complicated as I initially was led to believe, and that's fine by me :-)