Maximum Hack (Updated)
Apparently Crytek is into marketing and making money, are you surprised? Before Crysis 3 was released Crytek was trying to claim that you would only be able to run the game with a DirectX 11 capable video card. Similar story was pitched when the original Crysis was released. Crytek said that you would not be able to max out the game on a DirectX 9 card, so people thought they needed to buy a shiny new DirectX 10 card. Well this was not the case then and it is not the case now. If you were bummed out because you had an older graphics card then you are now in luck. Smart people of the internet were able to work around Crytek's marketing angle for Crysis and enabling highest settings for DirectX 9 capable cards with only few minor effects missing. And now the same thing has happened for Crysis 3. You can download the "hack" from here to start playing Crysis 3 on your older DirectX 10 card. So take that Crytek, you just got owned.
Update: Apparently it is not possible to download the files without going through some really big hoops which makes it look pretty shady. This might be a bogus claim in which case we are sorry if we provided you with false hope and wrong information.