What terminal emulators are available for heavy terminal users?

Besides GNU screen I would recommend tmux. It is a terminal multiplexer like GNU screen.

I personally use gnome-terminal as my terminal emulator. Using gnome-terminal, I run GNU Screen (byobu to be specific) in order to work with multiple windows. You can become really efficient with it. I use GNU Screen because I often have long-lived terminal sessions that I can check on or come back to later.

gnome-terminal running byobu (showing only one window open)

You seem more interested in a graphical solution for managing terminal windows. For this, I highly suggest the terminal emulator Terminator, which supports multiple windows without the need for a multiplexer like GNU Screen. You can split it horizontally and vertically as much as you like and there are even tabs. If you just feel like running lots of terminals, Terminator is really good.

Terminator (showing multiple Terminator windows open)

I like guake, which is a drop-down editor (named after the console in games like quake). You press a button and it slides down, press another button and it hides away. Press another button to make it full screen, etc. It supports transparency, tabs, color... you name it. (You could run GNU screen on the inside to get more than one visible at a time.)

It's on the top of this screenshot, transparently overlaid some stuff I was doing. (Click for full size.) I think it looks good.