The best guess is that it is going to depend on the computer. Carrying a DisplayPort signal through Thunderbolt is a feature but not necessarily a requirement for having a Thunderbolt port. But, yes, assuming that a given Thunderbolt port is carrying the DisplayPort signal, there should be no reason why the Thunderbolt display will not work on that port as a display, regardless of the kind of computer.
Now as for other features included, it may depend on drivers for those features and how the host operating system recognizes what other things are included in the Thunderbolt display. For example, how customized the drivers are for each extra included device in the Apple Thunderbolt display is probably yet to be determined. For example the speakers, extra USB/FireWire ports, network card, etc. may not work if the host OS can't find a driver.
The specifics of how a motherboard-based Thunderbolt port can talk to any given PCI-Express graphics card installed in that board is also an interesting detail to address, hopefully the next generation Mac Pro will give an example of this since its implementation is questioned as well in the Thunderbolt wiki article as well:
Because the PCIe bus does not carry video data, it is unclear whether a standalone PCIe card could offer a Thunderbolt port. The Intel Thunderbolt Technology Brief does not give a conclusive answer. Intel disclosed documentation where video stream is sent to a dual-Thunderbolt controller, with the video stream being only sent to one of the Thunderbolt ports, giving the assumption that video stream is not mandatory on Thunderbolt implementation.
It looks like many of questions will be answered in the future as more hardware is rolled out but here is a bit of speculation on the noted next generation Ivy Bridge Thunderbolt controllers.
Another aspect that makes the DSL3510 interesting is that it supports multiple internal DisplayPort inputs. What this means is that it could in theory interface with a discrete graphics card as well as the integrated graphics from an Intel CPU. This is likely to be the chip used by Apple in its desktop systems, whereas the more power efficient DSL3310 will end up in notebook products.