As of November 2013, Amtrak does not allow pets on trains (see Amtrak's website). However, there are bills in Congress which would require Amtrak to change this policy: H.R. 2066: Pets on Trains Act of 2013 and S. 1710: Pets on Trains Act of 2013. The bills state that Amtrak must designate at least one car (on all trains with two or more cars) for passengers with pets. The pets must ride in crates at all times and are only permitted on trips that are less than 8 hours. It also states that Amtrak should charge a fee to pet-carrying passengers that would at least recover the cost of implementing this policy.
You can follow the progress of the house bill or the senate bill on govtrack.us. The site also provides instructions for contacting your congressman about supporting this bill.
If you want to go even further, I've created a flyer you can print out and post at your train station, pet store, vet office, or kennel/day care.
March 2015 Update
The 2016-2019 Amtrak Funding Bill, H.R. 749, includes a provision directing Amtrak to introduce a pilot program allowing pets on certain routes. This bill has passed the house and is on to the Senate. The specifics are similar to what was proposed in H.R. 2066 and can be read in section 210 of the bill's text. The pilot program shall be in effect on the Northeast Corridor and routes less than 750 miles.
May 2016 Update
Amtrak has unveiled an official pet policy. Pets that are 20lbs or less are allowed on most trains on trips that are 7 hours or less for a fee of $26. For full details, see Amtrak's official rules.