In a semiconductor, why is it said that holes also diffuse?
"to say that the free electrons go to the p-type, leaving behind positive ions in the n-type and creating negative ions in the p-type?"

From your question what I understood is you think hole movement same as electrons movement in opposite direction. 

In macroscopic scale your thought might come true. But here we observe the movement of each electron microscopically so there is a little difference. Consider a P-N junction "electron movement" is defined as the movement of a single electron from negative terminal towards depletion region. In "hole movement" a single hole is moving from the positive terminal towards the depletion region. Large number of electrons will be participating in this movement. Therefore we cannot consider hole movement as opposite movement of electron.