Your screens may be old enough to have the old dirty look even when they are clean. Replacing the screen part may be the only way you can get rid of the dirty look. Wire screens will get this way, not sure about the newer non-wire type.

Another thought, did you use a cleaning solution of any kind or just use water on the screens in all attempts to clean them?

Lately I've been cleaning mine from the inside with a damp microfiber cloth. That seems to be getting a lot of dirt off very quickly and very easily. LOVE my microfiber cloths! I got THESE but you can find them in your local hardware or automotive store. They also clean the windows with only water. Super easy cleaning.