The "Bad Bunny" worm uses OpenOffice as the initial vector, which when an infected file is opened would run a script that did different things depending on which OS it was running on.
http://en.wikipedia.org/wiki/Linux_malware#Cross-platform_virusesA Windows based virus is a windows application at heart, and applications have operating system dependencies that mean, just like any other Windows application, cannot run on linux natively.
There have been some tests with various Windows viruses to see how well they work under WINE - the "emulator" used to run Windows apps under linux. With limited success (success being the virus working as designed).
Java applets are less likely to be an issue cross platform as java was designed from the ground up to be a virtual machine, so the underlying operating system is not exposed. Same with flash to an extent. However both have had vulnerabilities, and potentially be exploited in different ways on different OSs.