Do the teams in The Amazing Race already know which countries they will be visiting prior to the production?

Per this reddit AMA with a show contestant, the production team tried to throw contestants off the scent using four main techniques;

• Getting contestants to fill out visa forms for more countries than will be visited
• Getting contestants to fill out visa forms that are never even submitted
• Taking the contestants to countries where visas can be acquired on entry
• Taking contestants to countries where visas aren't required

Things may have changed, I'm apparently narcissistic because I stopped watching after the episode where we lost (actually it was just kind of painful to see all the stuff we could have done.)

The legs for our season were kind of planned last minute. We were originally supposed to start shooting in fall of 2001, but well, 9-11 happened so suddenly we spent half the season in Europe, even though we had visas for Egypt, Oman, Myanmar, and several other cool as hell middle eastern nations.

Since all the visa paperwork had to be done beforehand, they make you get visas for a dozen or so countries to try and throw you off. I was pretty confident we'd be going to Africa and the middle east, so the only jacket I packed was a hoodie (like a true Californian). Our first destination ended up being the Italian Alps.

this is backed up in a USATOday interview with the show's presenter; Phil Keoghan

Q: 'Race' shows useful travel survival skills, but it can make getting from country to country look way too easy. How do contestants have the proper visas if they don't find out where they're going until the last minute?

PK: Anybody who's selected sends a passport in, and we get visas. But we make it confusing. They get extra visas for countries they're not going to. Even if they get a visa for China, they don't know where in China they're going. And some visas are not necessarily given right away. Even the crews (accompanying contestants) don't know where they're going.

It's also worth pointing out that this is a TV show and not a documentary. That means that certain elements (for example long, boring sequences where people fill out forms and sit around in hotel lobbies waiting for visas to arrive) are simply edited out.