I don't resize the map though, and pixel rounding and linear only makes my character move veery fidgety while never really achieving anything in terms of the blurriness. I'll tel l the artist to change it to PNG-32 and see if anything changes.
What Drew and Paradox are saying is that sampling should be set to point, not linear. Pixel rounding isn't necessary if you have sampling set to point and is the likely the cause of movement looking fidgety.
If point sampling doesn't fix it, it could be an issue with your graphics drivers.