I used to ride Amtrak into and out of Chicago all the time with my Tikit. Here's what I learned.
1. ALWAYS bring the folding bike policy. Some conductors don't care about the folding bikes and will welcome you with open arms, but others can be pretty cantankerous about bikes. Be polite when showing them the policy and they'll let you on the train with a minimum of fuss. If you don't have the policy with you, you can expect that some conductors will give you a hassle or force you to buy a bike ticket.
2. There are three potential places to store your folding bike. 1) With the luggage, in the luggage rack. I usually put it in the lowest rack. 2) On some trains there are extra spaces for luggage behind a column of seats. 3) Despite what the policy states, some conductors will ask you to place your bike in the overhead.
3. If you have the opportunity to cover the bike, do so. Yes, it's an extra hassle, but it could help you avoid problems. That said, I never covered my Tikit and had very few problems.
One thing to note, most Amtrak are unfamiliar about the folding bike policy. When they give you a hard time about it, it's not because they just want to give you a hard time. There is a bike policy on Amtrak that requires that you purchase a bike ticket. This policy has been in place a lot longer than the folding bike policy. As Amtrak employees see more folding bikes, it will become less of an issue.
For now, be sure to carry a copy of the policy with you... and be nice to the conductor no matter what he/she says or how he/she says it. It's better to swallow your pride a little than to be stranded somewhere waiting for the next train.
--sam