My shower head is filthy with so much water scum and yucky stuff on it. Is there household products I can use that will remove it without the work of scrubbing?
Chlorine bleach works to whiten and brighten plastic shower heads.
Perhaps of related interest, apparently chlorine doesn't kill some types of microbes, but it will look clean.
Link to bacteria in showerheads article. The article suggests a few ideas, like replacing the showerhead entirely, which I don't consider a hack.
To get the built up residue off of your shower head, tie a bag filled with vinegar around the shower head, and leave it there overnight. It will clean everything off with no work involved.
CLR, vinegar (it can stain some finishes), bleach, lemon juice, or a baking soda + vinegar combo. Place your choice in a bag, attach over the showerhead, leave overnight, remove and run the water.
Usually I use citric acid for almost any obstinately pollution like limestone. It's very cheap and the low doses you use in household needs won't harm the environment.
Usage is simple:
Take a bucket.
Fill it with water.
Add about 1-3 teaspoon per liter, depending on how strong the pollution is.
Insert the dirty object.
Leave it be for some hours, e.g. if you're at work.
Avoid eyecontact! Citric acid is mildly irritating, so avoid long skincontact, better wear gloves. But you don't have to panic if you go something on your skin. Simply wash it away. Unless you keep your hand in the bucket for several minutes, you shouldn't notice anything.