Core 2 processors don't have the memory controller on the processor like AMD's Athlon and Intel i-series. The chipset on the motherboard has the memory controller. Find a motherboard that supports DDR3 for use with your processor.
The CPU does not have an internal DDR3 controller like the AMD AM3 CPUs or the newer i3/5/7. But you will find many crossover boards that have an added DDR3 controller. For example, I have an ASRock P43D1600Twins... It is a Core 2 board with 4 slots for DDR2, and 2 slots for DDR3. You cannot run DDR2 and DDR3 together, but you can run either of them individually.
But many LGA775 boards that do support DDR3 only have 2 slots and also usually have a 4GB or 8GB limit. My ASRock board can take 16GB of DDR2, but only 4GB of DDR3. So just read the small print on LGA775 boards that support DDR3 to make sure the controller supports the type and amount of DDR3 you want.
The Intel® Core™ 2 Quad Q8300 can support DDR3 when the board that supports it. The Intel Desktop board DP45SG will natively support DDR3 800/1066/1333 and will also support the Q8300 processor. I was able to find this board over on Amazon.com. I never heard of any issues with this board and the memory.