You can use the current version with other hardware, but it isn't trivial.
The LED Cinema Display connects to Macs using a Mini DisplayPort, which isn't popular on non-Apple hardware. However, you can get a converter like this one that lets you use the monitor with an HDMI connection. So, if the other machine has an HDMI out you could use that. I'm sure you can also find converters to use with VGA or DVI outputs.
The next version of the display will (almost certainly) use ThunderBolt instead of Mini DisplayPort. The physical connector is the same, but the data that travels over the cable is very different. These converters may or may not work with ThunderBolt; we'll have to wait for the product to come out and for someone to test.