Like it or not, today in the business world when someone sends you a document, it's usually going to be a Microsoft Office document: Word, Excel, PowerPoint. Luckily, it's not totally impossible at this point to read and write the Office file formats, so many alternative OSes have word processors and spreadsheet apps that will do a pretty good job of interoperating with the Microsoft Office-using world. Mac users can even use the real McCoy. The Mac version of Microsoft Office is, for now, still available and enjoys complete file compatibility with Office for Windows.
Microsoft is still in the driver's seat here, though. They control the Office file format, and they change it from time to time, making headaches for developers who want to build apps that are compatible. It's not outside the realm of possibility that someday Microsoft will design a new version of Office whose files are encrypted so only Microsoft applications can read them. With the DCMA, even the most rudimentary encryption would make it against the law to distribute an application that reads that format. Microsoft might not be able to get away with such a dastardly plan, but it illustrates the point that file exchange is a point of difficulty for the alternative OS. To be viable in the business world, every alternative OS must have office applications that speak Microsoft.
If your OS has a great word processor, it will probably also have a great file format that supports its great features. Maybe that file format will be based on a standard like XML, or maybe it won't. But one thing's for sure: if you want to send that file out into the world, you'll have to save it in Microsoft Word, RTF, or Text before you do, and when you get it back, unless the formatting and other features exist in Word or RTF, that formatting will be lost. Microsoft Word users do not have this problem. They can choose from the pantheon of outrageous features for their document, and inflict them on other users without fear. Most non-Microsoft word processors will be able to read the file, but probably don't support all the wacky features, so you may not see the file as the author intended. Now, users of older versions of Office also have this problem, so you're not alone on this one. People who do not upgrade their Microsoft software regularly are almost as much of an enemy to Microsoft as you are.
I should also mention filesystems here, as any alternative OS will also need to know how to read from and write to all the common Windows filesystems too. That has not posed a problem for the Users of major alternative OSes like Mac and Linux in a very long time, though.
Probably the thing that Microsoft's desktop monopoly gives them that makes alternative OS proponents the angriest is the ability to undermine standards. Let's use HTML as an example, as it is the standard that Microsoft is in the best position to control. It's also a good example because it's a very badly abused standard. Microsoft wasn't the first abuser either, Netscape was. The HTML standard simply evolved too slowly to satisfy web developers, who wanted to make pretty web sites, so Netscape developed non-standard tags to please developers. You can't really fault them too much. I remember those days. It was really hard to make a non-ugly web site back then. But the fact that sites looked better with the Netscape browser drove people to that browser. And eventually many of the Netscape-initiated tags made their way into the standard.
When Microsoft set its sights on the browser market, it took notice that aggressive innovation and a fast and loose outlook toward the standards could propel a browser's marketshare, and just as the early web era saw Netscape-only web sites, nowadays there are many sites that do not work well in any browser than Internet Explorer.
There have been various claims that Microsoft is in a position to undermine MPEG-4, DHCP, and even standards of its own creation like SMB. That Microsoft was attempting to undermine, or at least exercise some control over, the Java standard has been the subject of years of legal wrangling between Sun and Microsoft that continues to this day.
How this affects the alternative operating systems is clear: when standards evolve through neutral governing bodies, there's a level playing field. When one company starts to push the standard around, then everyone else has to play catch-up.
For those applications that you really need to run, many alternative operating systems have applications that allow some kind of Windows emulation. Many of these, like VirtualPC for Mac and VMWare or Win4Lin on Linux require an actual licensed copy of Windows that runs in parallel with your main OS. These applications work very well, though VirtualPC does not perform at full speed because it emulates the x86 on a PowerPC. Nevertheless, for the rinky-dink applications that we all occasionally run into they're a good solution, except for the fact that to use them legally you must pay for a license to Windows. (Note: VirtualPC does come with a license to Windows, though that license inflates the price of what normally would be a $50 application to $200)
There are other problems with this approach. Whatever advantages of usability, performance, stability, or interoperability that attracts you to your alternative OS is counteracted completely when you use an application in this manner. You have to "leave" your own environment and temporarily inhabit the Windows world. And lastly, it's an admission of failure. It's tangible proof that you still need the mainstream OS to do your thing.
The Wine project on Linux (and Unix) attempts to implement the Windows API without Windows, but it won't run all Windows software. So it's a really exciting solution for those applications that it supports, but it's no panacea for the software problem.
Running Windows-in-a-window is a handy option for the alternative OS user, but it's sidestepping the issue.