The RIGHT i3, CAN work ok with a high enough end GPU to have a good experience. They have some strengths and weaknesses. Models with a fast enough base clock have fared well against FX in SOME areas. Even against the i5 in SOME areas. That I can agree with. But to flatly say that an i3 would outperform FX 8 across the board is not accurate and in consideration of budgets, is NOT always clear regarding what to choose. Sorry DC, had to pick on you there.
Lower end i3's don't do well at all against the i5 or FX 8, but some of the higher end models do, but the're so expensive that you're often better off just adding a few more bucks to the pot and getting an i5 or FX 8 core and giving it a nice overclock. My 8320 at 4.5Ghz paired with a 290x doesn't struggle at all and I rarely encounter ANY issues that I suspect are CPU related. There are a few titles that don't like it much but overall it does well.
If you're even "thinking" about an i7, that means budget isn't much of an issue and I'd really recommend going with either a 4690k or Xeon E3 1231v3 if gaming is to be your main focus and you don't feel you need an i7. With the same single core performance as a similarly clocked i7 and the same 8 threads, but priced similar to the i5 since it can't be overclocked and has no integrated graphics, it makes a good halfway point choice between the i5 and i7.