Presently, pets are not permitted to travel on Amtrak trains unless they are medically certified service animals. But that may change if a Federal bill just introduced in the house moves forward.
H.R. 2066, the Pets on Trains Act of 2013, introduced by U.S. Reps. Jeff Denham, (R-CA) and Steve Cohen, (D-TN), would require Amtrak, the national rail operator, to implement a pet policy to allow passengers to travel with domesticated cats and dogs on certain trains. Should the bill pass, there would be fees involved as well as a mileage limit. Additionally, pets would have to travel in carriers or crates.
Amtrak always defended its pet policy by citing that it was an interstate service in the US and it would be complicated to adhere to differing state legislation. However, Federal legislation may change all that. Hopefully, it will get some support. Pets can travel in trains commonly in Europe (not Eurostar, however). Why can we not change this pet policy and provide this service in the US as well?