Not cross country, but we did do Amtrak from VA to FL (auto train) with 5 boys under the age of 6. Sitting in seats, no sleeper. I'm not sure I'd do it again unless my kids were older...I would opt to fly over taking the train if costs were the same. The seats aren't bad, but the kids had a hard time sleeping in them. While they do recline, they don't go horizontal. We brought pillows/blankets etc. We were on the lower level of the train and had a section to ourselves, thank goodness. (Lots of elderly passengers who would not have enjoyed our crew.) The food was pretty good and the kids LOVED eating on a train. They loved the bathrooms, go figure, and it was nice to be able to walk around and not have them strapped into a car seat/seat belt. That said, it was LONG, and ours was only overnight (4pm to 8am the next morning.)
Sorry I can't be more helpful. If nothing else, it would be an adventure and your kids would probably talk about it for years.