I read that DTS packages aren't supported in SQL Server 2012. I know there's a Backward Compatibility package/option out there that we used for SQL Server 2008R2 to run DTS packages. Will that Backwards compatibility package not work for SQL Server 2012?
For 2008 R2 http://msdn.microsoft.com/en-us/library/bb500440(v=sql.105).aspx
As the documentation says, support for "migrating or running" DTS packages was completely discontinued in SQL Server 2012. Even if you convince your customers to move to SQL 2012, you'll have to upgrade the packages to 2008 first and then upgrade again to 2012, so the sooner they agree to upgrade to some newer version the better, although I appreciate that - rightly or wrongly - many customers will not see the point in paying to upgrade something that still works.