So I just purchased a new computer and moved my power supply and graphic card from my old Core 2 Quad Core desktop to my new one:

Intel i7-3770 @3.4Ghz
8GB DDR3
Radeon HD 5750
1680x1050 resolution
Win 7 ultimate
200GB free space
*With latest Witcher 2 patch installed and latest ATI drivers

I used to just be able to handle this game on my old quad core on low settings and considering this PC demolishes my old PC in speed I'm not sure what the problem is. Even on medium graphics the game's frame-rate hiccups. Could it be my graphic card is too old?

CPU is not that important for frame rate, graphic card on the other hand is the major contributor here. I'm not too familiar with Radeon lineup of cards but it might be the bottleneck for you if nothing else

It could be the GPU. The 5750 only has a 128bit memory bus. The witcher 2 really hungers for a lot of memory bandwith. You could try playing at a lower resolution. If the game runs good at medium settings at for example 1280x800, it could be that the limited memory bandwith of your videocard is the cause of low performance.

yes, your graphics card is the problem.

I only have an Intel i3 CPU but with my Sapphire Radeon HD 6870 I can run the game in 1920x1080 on high settings.

I wish I had that CPU, but your GPU is old and weak (it was pretty weak even when it was released, even my 4890 is better).
I suggest you to buy an AMD 7870 or a NVIDIA 660Ti. Anything worse will limit your powerful i7.