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Mass storage will work on Linux, but internal memory will not be detected. There are work arounds. Take a look at these pages:
http://talk.sonymobile.com/t5/Xperia-Z1/Using-with-Linux-to-copy-files/m-p/425669#M4531
http://talk.sonymobile.com/t5/Xperia-Z1/Sony-PC-Companion-for-Ubuntu/td-p/437965
http://askubuntu.com/questions/199280/how-to-install-pc-companion-software-of-sony-xperia