It shouldn't harm your CPU to try right away, but a BIOS update (which should be harmless) should improve your chances of your CPU working. You might not have an EZ flash option on your motherboard, but updating your BIOS isn't impossible. Check your motherboard manual to see your options. Which exact motherboard do you have?
Also, there's no guarantee your the needed BIOS version that supports the 5600G is already installed simply cause you bought it after a certain date. There's a chance that your motherboard was already produced and on a shelf before the BIOS version was released.