edit In retrospect, I'm not really sure that I've answered your question, since lmms is not necessarily any simpler than your windows software, and I'm not aware of anything that's very simple to install that will just give you a piano sound without some configuration. But if you end up needing to learn more about this stuff in order to get what you want, this may prove a useful starting point.
Under linux or windows, you can use lmms: this is a free-as-in-speech "digital audio workstation" (DAW). It's a bit more than just a way to play piano sounds, but is fairly easy to set up and use. The getting started guide is a good place to, er, start. If you're very new to this, it's probably worth spending an hour or so reading through the manual. There are some other guides, on installation for example, at their main wiki page.
I'm not aware of anything under windows that's more specifically geared to the task of playing piano sounds; in general this is done by DAWs, which provide a management interface for hooking up various sound generators to make piano sounds, drumkit sounds, vintage synth sounds, never-before-heard-sounds, and many more. They also provide the ability to sequence (program) the playing of samples at particular pitches, volumes, etc., to create melodies, rhythms, and entire pieces of music that can be played back and edited. You can also layer effects on top of your tracks, generate a musical score, and do other interesting things.
I personally prefer rosegarden to lmms as a DAW under linux, but I think it's a bit less beginner-friendly; for instance, it doesn't include any sound generator plugins like qsynth ("synths"), but expects you to install them and link it up with them. lmms includes a couple, so it lowers the learning curve a bit. One advantage of rosegarden that may interest you as a piano player is the option to program music by scoring it using traditional staff notation. I'm not sure if lmms provides this facility. Note that rosegarden is only available for linux.
Under linux, there are more specifically-targeted programs, like qsynth, that do one thing and do it well; in qsynth's case, this task is to play sampled sounds at various pitches, including, IIRC, some decently realistic piano sounds (lmms I think has one or two of these as well). However, because programs like qsynth are built as components rather than an applications, you'll need some understanding of the infrastructure in order to use them. DAWs like lmms or rosegarden are designed to help manage that infrastructure, but if you want a more stripped-down system, you can run the synth programs without a DAW.
In addition, the JACK Audio Connection Kit is required by many linux music production programs (lmms excluded) JACK is a sound server like ALSA (linux's default sound server) that is very low-latency and provides, via
qjackctl, an interface to hook synths up to MIDI inputs (like your keyboard) and sound outputs (like your soundcard). If you have JACK installed under linux, you can use qsynth directly with it, without needing to use a DAW. Installing it under most distros is mostly just a matter of installing the package, though you may have to do a bit of delving to get it running entirely smoothly. You have to start it up before you can run it; this is also done with the GUI app
qjackctl. Depending on your distro, you might need to install the package
qjackctl in addition to the
jackd package. These package names are themselves specific to the ubuntu distribution, but common among some others in the debian family.
Again, because LMMS can use ALSA drivers interchangeably with JACK drivers, JACK is not a requirement to use it. However, when using ALSA drivers, the latency will be higher. Low latency is important because it means that sounds come out of the speakers as you play the keyboard, rather than a noticeable amount of time afterwards. Annoying latency is maybe about a tenth of a second; some systems under Windows or using ALSA will stick you with twice that or more. JACK will get you very low latency sound, to the point where it's not noticeable at all. This is possible under Windows as well, but you may have to tune your system or use a different sound card or driver.
If you're considering installing linux specifically in order to do music production, you may be well off to use a distro geared towards that. This will take care of the details of setting up JACK, installing a DAW, etc., and provide you with a slew of synths and other related apps to explore. I've found ubuntu studio to be pretty solid; there are some other options listed in the "External links" section of its wikipedia entry. If you really just want to use your keyboard as a piano, you may find the volume of options it provides overwhelming, but if you want to explore more possibilities later, it will make that very convenient.
Finally, if you do have trouble getting things working under linux, the virtual keyboard plugin can be helpful to figure out if the problem is with your software setup or with the interface with your keyboard. This will be installed with ubuntu studio, and probably with the other multimedia-oriented distros; otherwise you can install that package to get it.