Copying TeX packages from MiKTeX Windows to a Linux TeX Live machine

If your distribution have the actual TeXLive 2012 (as Debian testing, for example) you can do a full installation with the official packages, as well as remove all the texlive packages and make a fresh installation in /usr/local following one of the options explained in TeX Users Group web site since this allow and install/update TeX packages through their own package and configuration manager (tlmgr) instead of apt-get or the package manager of your distribution (rpm, etc).

If you distribution still have an outdated version as probably TeXLive 2009 in Ubuntu, without tlmgr, is better the second option, since there are significant changes and improvements. (some interesting examples that you can find in this site cannot be compiled with that version).

Mixing files with those of a MiKTeX installation have every chance of be a source of problems. As in this answer said, install extra packages locally (~/texmf), always according to the readme file, and when you checked in CTAN (for example) that the package is not in TeXLive and therefore you cannot install with the package manager of Linux (as apt-get) or TeXLive (tlmgr).

To say it short: Don't do it!

TeXLive and MiKTeX are different TeX distributions with own, different package manager. So install both distributions with their installation program. For example TeXLive uses pearl that MikTeX can not use in the same way ...

If you want to use special packages on both distributions you can create a local texmf tree, install the packages (following the readme files) in your local texmf tree and use this local texmf tree in both distributions.