My eyes are bad infrequently now, but when they are, I dampen a washcloth and heat it up in the microwave for 7 or 8 seconds to lay across my eyes. This warms the oil in the meibomian glands along our lash lines. I have to reheat it several times during the 5 minutes or so I do this, so a rice baggie sounds like a great idea. An old white sock filled with rice would work, too.
After I've warmed the area, I use my index finger--clean, of course--to stroke close to the upper lashes, starting close to my nose. After a couple of minutes of stroking, I use Celluvisc drops (the thick ones) and my eyes feel better. I've never massaged along the lower lid.
My opthalmologist says to think of the three layers of tears as being like salad dressing; free-flowing oil is important in keeping your eyes comfortable.