Is there a green cleaner that is an alternative to Woolite?

There are quite a few! I know my local supermarket carries Seventh Generation products so that is my first suggestion.
http://www.drugstore.com/products/prod.asp?pid=173017&catid=198307
Here is another brand called Planet Delicate.
http://www.drugstore.com/planet-delicate-laundry-wash-hypo-allergenic/qxp137788
They have good reviews on this website, but I haven’t tried either of them. I have used Seventh Generation toilet bowl cleaner though and it’s good.
I also use shampoo sometimes. Anything delicate enough for your hair is good for fine washables. Just stay away from moisturizing shampoos because they may leave a residue.

Years ago someone recommended Ivory dishwashing soap for hand washing delicates.

Yes, Ivory soap is good too.
Has anyone ever tried this soap?
http://www.zote.com.mx/english/about.htm

It’s called Zote and it’s made traditionally in Mexico since 1970. The website says it is safe for delicates but effective on stains. It says it is good for baby clothes too. I’m curious if anyone has tried it. Apparently some people swear by it.
Here’s a testimonial:
http://dulcecandy.com/2009/04/mexican-soap-zote-for-your-delicate.html

Pink Solution I use it on everything. It is an awesome cleaner and very gentle. One bucket lasts for years.

Can I use my regular liquid laundry detergent for hand washing delicates?

I wouldn’t. You can buy Dr. Bonner’s mysterious elixir at any health food store worth its salt.

Dr.Bronner

Or the lavender which I like

I wouldn’t use regular liquid laundry detergent even for regular laundry any more.