This tutorial assumes you’re running on Ubuntu with Unity desktop and ready to go.
First, you need a 3D video driver. Search “additional drivers” in Dash home. In pop-up “Additional Drivers” window, select an appropriate driver and then click on “Activate”. Enter you user password and it starts downloading and installing the driver. You’ll probably have to reboot after that.
Second, install Java. Launch “Ubuntu Software Center”. Search java, then choose to install “OpenJDK6 Java Runtime”(not seven).
Now, go to minecraft.net and download the minecraft.jar file. Open it via archive manager and extract favicon.png under net/minecraft/.
Go back to minecraft.jar, right click to go into its Properties window. In Basic tab, click on icon and change it to favicon.png (the picture file you just extracted). In Permission tab, check “allow executing this file as a program”. In Open With tab, choose “OpenJDK Java6 Runtime” and set to default.
That’s it! Run minecraft.jar, it’ll download what it needs, and go.