There's some applications which support minimizing to a tray icon natively. There are others that can be extended to do this. What I'm looking for is something that will allow me to set rules on which windows should be sent to the system tray when I minimize them.
On MS Windows, I have a program called DexPot that allows me to do this. You can tell it, for example, that if a process is called "putty.exe", when it minimizes, it should be sent to the tray, rather than taking up a tile on the taskbar.
I've found a program called AllTray, but that just seems to let you pick a currently open window to send to the tray, rather than creating rules like "Always minimize terminal to the tray".