In NOL you can check your bag by showing your ticket/reservation. There may be a fee if you want to get a carry on bag back from the baggage room.
In Chicago there are pay lockers that accept quarters or credit card. For security in CHI, you give a finger or thumbprint scan. You have to give the code from your claim check and the same print to get your bags out of the locker.
Washington has a private contractor that seems to charge more than SmarteCarte. I only used this once and felt it was a rip-off IMHO.
If you arrive in Washington on the train you have about 48 hours to pick up your checked bags.
Are you traveling in coach or a Sleeper? If you are boarding a sleeper, you can leave your luggage in the first class lounge in Chicago, New Orleans or Washington.
This isn't related to the cities you asked about, but I discovered that for $4 each suitcase my friend and I were able to leave our luggage at Saint Louis as well as at Los Angeles for a few hours. (The agent at Los Angeles even offered to store our backpacks for free, but we carried them with us during the day as we did some sightseeing.)
I think someone else mentioned staffed Amtrak stations storing bags during the day for $4 - I didn't realize this before our trip but it was very handy to give us some time to explore the cities before departing on the train. We had ticketed sleeper reservations so I don't know if that is a requirement for day storage. We were going to ask a local hotel if they would do the same for us but Amtrak was great - very convenient.
The nice folks at ROC stored our bags when we went to dinner during the interminable wait :angry2: from the Maple Leaf to the LSL. No Charge - they also checked our BIG bag to STL even though we were only going to ALN :wub: