When you power on the phone, does the screen come on at all and show the Samsung logo? If so then it is likely to be a software or firmware problem, and a factory reset might fix it. If there is nothing at all during power up then my research into my own "blank screen" suggests it is hardware.
Did it sustain any sort of physical impact, like dropping on the floor? If so, then there is a chance that the main pop connector has come loose, as the answer above suggests. If it has sustained environmental damage (water, heat etc) then it is more likely that one or more components have failed.
I've seen a lot of posts on various android and samsung forums that suggest replacing the lcd/digitizer assembly, but all of these have described the soft keys (back and menu) lighting up during power on. In the case of my "blank screen", I don't get anything at all on startup, not even the soft keys light up and I have confirmed my screen *does* work in a different phone, so the fact your screen is blank does not necessarily mean the screen is faulty. A replacement screen is cheaper and easier to source than a replacement main board - but if you guess wrong the first time and end up buying both then you have probably spent more than a new phone or service-centre repair. So my advice is not to rush into anything until you can narrow down the issue with some degree of confidence.
There are hundreds of threads on the Internet covering samsung s3 "blank screen" problems and many have solutions. You would be well advised to search through some of these to see if you can find one where the symptoms (and possible lead up events) are close or identical to yours. Without some further information it would be impossible (or shear luck) for anyone to give you a definitive cause and solution.