Unix systems might not have an exact equivalent because HyperTerminal does two unrelated jobs which in unix are done by separate programs.
One job is providing a text terminal inside a GUI environment. This job is done by terminal emulators such as gnome-terminal, konsole, rxvt, Terminal.app, xterm, ...
Another job is converting between user interactions and serial port interactions (since you use HyperTerminal to connect to it under Windows, I presume the device uses the serial-over-USB protocol). Under Linux, minicom is often used for this; there are undoubtedly alternatives. Whatever program you choose needs to be connected to
/dev/ttyUSB1, etc, if you have more than one device).