I too tried one of the new Fire TV devices and experienced the stuttering issue, but not just on Tablo. I went back to an older 2nd Gen Fire TV that hadn't been used for several months, and it was running an older software version. At first it seemed to work ok, and then it updated to the same software version that the new 3rd Gen Fire TV was running, and, the stuttering started again. So, it may have something to do with the latest Fire TV software version. Also, I have a new Samsung TV with the Amazon Video built in, and it has the same stuttering issue. Hopefully, it will get resolved soon.
Also, on my setup, the Tablo playback quality is not nearly as good as it used to be with the older Fire tV product. It looks more like SD tv rather that HD. I also seems to vary during playback. Frustrating. Still waiting for Multi-Channel audio on the Tablo