I was in this position 2.5 years ago when I started with my current employer. Being that we are a non-profit it was placed upon me to cut costs as seamlessly as possible in a fashion that would have as little effect on the users as possible.
My first thought was to go the route that you are thinking. Then I got to know my fellow employees (ie. customers). As stated in previous posts most didn't know how to even follow simple directions to add a network share drive let alone learn a new OS without being bugged with the simple, mundane questions every .005 seconds.
What I've found is that most users don't like change. They get used to things and like them to stay the way they are and then you hear nothing but complaints and "how do I..." for weeks or months even though you have provided the simple documentation on how to do something. A great example of this is the way that I've had my users react when I upgraded from Office XP to Office 2k7. Some love it, most hate it and I am still dealing with some people on it months later!
Another thing, are you running proprietary applications other than Office in your network? This was probably the biggest deciding factor for me not going this route as we do have some proprietary software that would just not work correctly under WINE no matter how hard I tried. Maybe things have changed now...
My point is, not only do you have to take your boss into consideration when making this change (as ultimately it's their call), you have got to take your fellow employees into consideration as well. Initial cost savings in going a route like this may be great off the top, but what are the hidden costs going to be? What's the cost going to be of you having to do all the re-training on using Linux instead of Windows? How many hours will this take you to implement and then what's the cost of that? What's the cost going to be in you running around doing the simple, mundane tasks that your users won't do (and trust me they won't) because they don't know how and won't read your simple guide on how to add a network printer under Linux even though the one they should be using is scripted to logon, but they insist on having this particular one available to them?
Remember the KISS (Keep It Simple Stupid) rule. It's not only for you but your "customers" as well, and often one will affect the other in your happiness.