I had a friend with an identical problem, running an ancient single-core AMD (I forget the model, but it was like 2.2GHz). Performance wasn't great but it was still playable - he got like 20-50 fps on 4x4 maps, depending on the number of dwarves - except occasionally the program would freeze, that is the interface and everything would stop responding completely, for a couple seconds before resuming with a rush.
edit: I did some testing, my old 2.4GHz Core 2 Duo gets the same problem; it runs a fresh 6x6 embark at 160-200fps, clearly the hardware is powerful enough. And yet I get half-second freezes now and then, a few times a minute, makes the game frustrating to play. I never saw this before because I used to limit myself to 4x4 on that machine.
A little further testing: my i7 runs a 6x6 smoothly, but I can get it to exhibit the stuttering on a 12x12. FPS is a slow but playable ~25, however there are moments of very brief freezing. It's more noticeable if you watch some dwarves covering a long distance, they'll move steadily for a while, then they'll all seem to pause for a moment, then quickly rush over 5-6 steps, etc.
So I guess the short answer is: get better hardware if you want to run a 6x6 smoothly. The long answer is: there's some limit to the embark size your computer can handle depending on its specs, past that point you'll experience brief freezes even if your fps is otherwise good. As to why that happens, well, one might guess that DF is occasionally performing a surge of calculations that will freeze the program until it resolves, but the freeze is only long enough to be noticeable if the ratio of your embark size to hardware specs is high enough.