Here in Vancouver, WA we are not only blessed with green outside of the house (moss) but inside as well (hard water). Almost every sink, shower and toilet we have has green stains in it. When we moved in to the house, we thought perhaps the previous owners were using the tub to die things green. But I did some Internet searching and found out they were mineral deposits.
So, I tried the gentle approach: soap & water, vinegar, baking soda, but nothing could remove the stains. I tried Oxi Clean, I tried bleach, and then I tried CLR. While CLR did make a dent on some of the stains, it was not a slam dunk. What finally turned our toilet from nasty green and brown to a sparkling white was a pumice stone and about 2 hours of elbow grease. I used the bathroomstone by Earthstone, it was only about $4, but it did such a great job I would have paid $100.
As for outside, I scraped up the big moss with a paint stirrer, then swept the concrete patio. I applied Oxi Clean and water to the patio, brushed the solution around, let it sit for an hour, then hosed it off. It certainly made a difference, but it didn't make it all go away. So I had to use chlorine bleach mixed with water on the trouble spots.
All in all, I'm happy with the results, and I'm happy to know what actually works!