It is still possible (and in my experience not that difficult) to get FAA tower tours (and sometimes even TRACON or Center tours), though you should be reasonable in your expectations here -- You are probably NOT going to get a tour of the tower at JFK or LAX.
The FAA used to actively encourage pilots with an instrument rating (or pursuing instrument training) to sign up for ATC facility tours under a program called "Operation Raincheck" -- This program is still running, though it's been substantially scaled back (Poke around on the FAA Safety Team's SPANS system, or sign up for notices in your area and keep a sharp eye out. Tours usually have a very limited number of slots available, and they fill up fast!).
(As an example of how quickly the slots fill up, when N90 (New York TRACON) last had a Raincheck event it was full within a few hours of being announced. No, I didn't get a slot.)
At towered GA airports - particularly ones with a lot of flight training - you can also contact the tower directly. The printed AOPA Airport Directory has facility numbers in it but that's a little out of date. You can also get the phone number from Flight Service, the local FSDO, or ask the tower/ground controller for the number if they're not too busy.
When you call the facility explain that you're a pilot/student pilot and would like to arrange to tour the facility (and if possible shadow a controller). They will usually ask for your name, address, airman certificate number, and a few other things to run a check and get approval to let you in to the facility. You will also be expected to have government issued photo ID with you when you show up.
My experience arranging a tour of our local tower was extremely pleasant - they are happy to do the tours and let you shadow a controller, and I even got to pick their brains about some peculiarities in local procedures.
It's a great educational experience, and I would recommend it for any pilot/student pilot.