So, after some more digging this morning and banging my head I found a solution using SPServices.  First, convert the field from a complex to a simple drop down:

    $().SPServices.SPComplexToSimpleDropdown({	
        columnName: "myColumn"
    });

Then, you can get the ID (and then the value) by using what SPServices has cleverly injected:

    $('#SPComplexToSimpleDropdown_myColumn').val();

I'm still not sure of a way to do this without SPServices but I think it has to do with how SharePoint is doing funny business with the `input` field.