I guess I'm confused then, I though Rodi was a BT program.
No, Rodi isn't a BitTorrent client or server. I think it's a hype to say "torrent" to every program with multisource downloads...

I was considering using Mute. Is that a BT program?
AFAIK the developer of Mute will integrate multisource download soon. Mute isn't a BitTorrent program, neither.
Are there any anonymous BT programs then? Or am I stuck using BitTorrent or Azureus?
For downloading a regular torrent on the Internet you need a program which has implemented the BitTorrent protocol. You named BitTorrent (the original client) and Azureus. But there are
many other BitTorrent clients.
Downloading regular torrents in an anonymous way is possible. You have to tunnel the data traffic of a BitTorrent client through anonymous networks into the Internet, e.g. with Tor or I2P. But this is slow and those networks aren't made for huge filesharing activity.
Anonymous torrents are possible on the anonymous P2P network "I2P". The tracker and the BitTorrent clients transfer all data over this anonymous network. You can't download I2P-torrents on the Internet! Downloading regular torrents on I2P itself isn't possible either!
Greetings,
Nemo.