It turns out that there's something ugly that Microsoft did to the MDAC that messed things up pretty thoroughly. I grabbed the newer versions of the MDAC drivers (version 2.8, so that we're backward compatible to Windows Server 2003) and that fixed up the connectivity problems.
Unfortunately, it just looks like the SQL statements that we've written to pass to the Oracle DB are wrong. And I don't speak SQL of any flavor.
So I have called for help.
And that is quite enough for a Friday night.