The $459 rail pass mentioned in the article seems like a good option (15 days, 8 stops), but you have to be in love with the idea of taking the train.
A lot of American friends were surprised we were doing this - they had this idea that it would be uncomfortable and boring. They just couldn't have been more wrong. Between the comfortable room-ette things, often very friendly and varied fellow travellers, surprisingly tasty food and absolutely amazing scenery, it's a fond memory.
Of course, we scheduled in plenty of time - if you need to be somewhere punctually, Amtrak is not really your best bet.