How come my dishwasher leaves grime on dishes?
This is (most likely) a hard water issue. There are trace levels of minerals (calcium) in the water and when it dries the minerals are left behind. This leaves a white film that is not dangerous but is unsightly. 

There are commercial products that will help with this. 'Jet Dry' is the most common of these and I have found that it works quite well. There are competitive brands out there which report to do the same. 

You may also need to "clean your dishwasher" (sounds odd, I know). Again I point you towards a commercially available product, CLR. Basically you put some CLR in your empty dishwasher and run it. The exact instructions are on the bottle. 

(This is not a paid endorsement of either Jet Dry or CLR)