A good power washer is the only thing I know of. Don't waste your time with a piddly one -it'll break your back.

Clorox....either straight or mixed 50% with water...but it will take longer.
Pour it on....wait and flush with lots of water.
I did it for years when I had a concrete patio....and so did the woman who lived here before me.
Linda C

Thanks for the tips! DH bought a power washer a few years ago, but it isn't real effective. It had a really narrow stream, so you could practically "draw" on the concrete, but to get it clean all over was impossible!

Question on clorox -- how diluted does it need to be to not kill anything?? Will the 50/50 solution be okay?

You may want to check into renting a more powerful power washer. When I had my house painted, I paid a little extra and asked the painters to power wash my sidewalks, concrete back steps and driveway. I swear, it was like night and day difference. And, it didn't take them that long either. The sidewalks had bad staining from hickory and oak trees. The back steps had some mold and were very dark grey. After power washing they were practically new looking. I had no idea they would come up looking that good.
I would be concerned about all that bleach for safety reasons, myself. Then again, I have a garden right next to my patio so I wouldn't want to kill anything it it.

You were probably using the wrong attachment on the power washer if all you had was a stream. With the proper head on it, a power washer will make your concrete look like new, with little actual work.