The two big questions that have come out of this challenge have been the logistics of effectively air drying laundry as well as how to keep your clothes from getting crunchy.
A lot of these suggestions were made in the comments of other posts, but I thought I'd put them all together in one post for people who are looking to try a solution to the problem.
1. Look at your detergent - If you are using a petroleum based detergent without any kind of fabric softener then it's possible that it's leaving more of a crispy film on it. Try a gentler soap. Regardless of the kind of detergent you are using, the other possible issue could be that you are using too much detergent and not all of it is getting rinsed out.
2. Add a softener - There are natural fabric softeners available, but the most commonly used, and cheap, one is vinegar. Add it to the fabric softener cup or to the load directly as a rinse if you don't have a newer machine.
3. Spring for a little drying time - Some people swear that drying their laundry for 5 minutes before hanging them out to dry helps keep them from being crunchy. I prefer finishing the laundry in the dryer for two reasons - the first is to make sure they are dry (which is hard to do here in rain and humid city) and the second is because it takes out the crunch. If they are already dry, you can just run them on "fluff".
4. Give it a good shake - Sometimes just shaking your clothes out really well before hanging and/or drying them where it's windy so the clothes move around can help prevent stiffness.
Did I miss anything? Any other suggestions?