I downloaded your image and opened it in Photoshop—most of its colors are outside of the typical printing gamut. Go to View > Gamut Warning. All of the areas that turn gray in that view are colors that a printer cannot reproduce. Turn that off, and go to Image > Mode > CMYK Color. A warning dialogue will pop up telling you that you're about to change the color profile from sRGB to, probably, U.S. Web Coated. Go ahead and hit "OK." You'll see the colors immediately change, moving into a printer-safe gamut. If you go back to the gamut warning, you will see that none of the image goes gray any more.

In order to get those luminous greens, you would probably have to do a lot of experimenting with different papers and a very expensive printer, the sort that professional photographers use. Even then, you'll never get quite that bright a color, I don't think.

One other thing to think about is that it is nearly impossible to set up a monitor to represent colors exactly as they will print unless you have use a calibration device to profile both your monitor and your printer. You can get pretty close, though. Monitor calibration and gamma.