Clothes that are dry clean only drive me crazy. I live at the cleaners every week and I hate the bill every time I walk out of there. I have, however, learned that a good cleaners is worth every penny (this was after one of those cleaners where everything is $1.99 ruined some J.Crew).
Anyway I took a leap of faith and used the handwash/delicate setting on our washer and then laid the Lilly Elsa shirts flat to dry.
Then I used my new steamer. You could also try ironing your shirt with a hand towel between the iron and your shirt.
As always, try a tester spot in a not very obvious part of your shirt before you go to town. That way if it messes up your shirt, it isn't totally out of commission. I also take no responsibility if you try this wash/steam method and it doesn't work for you! ;) Go to the cleaners if you have to but whatever you do, take very good care of your silk shirts and dresses!