I'll take a first stab at this. [Well, when I started writing it, it would have been the first stab. I'm a slow typist.] Others will no doubt expand on it. There are some similarities between high ISO problems with film and with digital. With film, the higher the ASA (ISO) rating, the bigger the silver grains in the film emulsion. The bigger the grains, the less resolution you get. Images can look 'grainy'. This is more noticeable the bigger a print is. A lot has to do with film brand and how it's processed. For me, I could shoot Ilford HP5 (ASA 400) and push process it to ASA 800, and still get finer grain than Kodak Tri-X (ASA 400) processed at 400. That was a consistent result, not a one off. Of course, processed at 400 the HP5 looked even better.
With digital sensors it's not grain size but electronic noise. Crank up a cheap audio amplifier and you'll hear hiss in the background. Even expensive amplifiers may have some at extreme volumes. It's the nature of the beast. Cranking up the ISO on a digital camera has the same effect. It creates image 'hiss' (noise) that shows up in pictures. How much of this you deem acceptable is based on individual tastes. It does affect resolution (sharpness) and can affect colors as well. This is referred to as luminance noise and chroma noise.
In my case, I found that if I turned up ISO beyond 400 on my K10D, the amount of image noise was unacceptable to me. On my K-5IIs, I'm good up to ISO 1600. I have my KP set to max out at ISO 6400 right now. So as sensors and camera image processors improve, the 'useable' range of ISOs gets larger for a lot of people. Some cameras have built-in, adjustable noise reduction capabilities. There is also computer software for fixing (or at least improving) image noise. Topaz De-Noise is just one example.
Besides image noise, a general rule of thumb is that as ISO goes up, dynamic range goes down. This is the camera's ability to capture detail in the lightest and darkest areas of an image. Decreasing dynamic range can be considered to be decreasing image quality, depending on the subject. Some cameras have settings for highlight and shadow correction which may help to some extent.
The bottom line - shoot at as low an ISO as the scene will allow to get the best image quality. What is an 'acceptable' ISO limit for the K-3 III? Hundreds of opinions to follow.