Linen & cotton tend to hold wrinkles. Aside from shaking my laundry, I also rub (smooth) the wrinkles out with my hands while wet & you'd be surprised how this helps. Also give them another good shaking after they are dry.
The more cotton you have in your clothes, the more wrinkled there will be. I don't like using the dryer for more than I have to. When left to dry naturally, they are softer, smell better and last longer. BUT, the wrinkles! What I do is put everything in the dryer for about 10-15 mins and then immediately hang up.
I shake them out and straighten the collars and what not. Then hang around the house or outside in the shade. I have done this for over 30 yrs. I hate to iron clothes, when I was a kid, it was my JOB to iron the family' clothes, so I really HATE it. If things need ironing, they don't get worn again. This way of doing it, works really well.
Also, if something like rayon or whatever comes out really wrinkled from the last spin in the washer & it's advisable not to iron it at all, i put it thru the normal wash (with luke warm water) and on the last rinse, I don't let it do the final spin. I take it out of the washer dripping wet, hang up and smooth out the material and over the next couple of hrs, I keep straightening it, till the front and back of the garment doesn't stick together.
This may be too simple of a suggestion, but if there is no breeze on the day you hang them out, they'll stay wrinkled and sometimes stiffer.
Maybe you could try using the slowest spin speed on your washer. This way it doesn't press the clothes too tightly together when spinning. It may take a little longer to dry them but I don't think they will be as wrinkled. Also, I'm sure you probably do this but shake each item before hanging on the line. Hope this helps.