Hi Amanda
I just noticed this thread by accident since before I thought the title referred to a lily. If you are referring to the white flowered Spathiphyllum it should be similar to the anthurium I am growing since it is also an aroid. I just posted it.
Anthurium crystallinum in water culture
My experience has been that the roots of plants that previously grown in potting soil, bark etc, come with rot producing bacteria and fungi that thrive best in the juncture between water and air since they seem to require wet and dry cycles to reproduce. For this reason I found that new roots and rhizomes grown completely under water tend to be free of these and as soon as new growths and roots are strong enough I remove all old roots as soon as they deteriorate and parts of the plant that contain any signs of rot that penetrated deep so it cannot be cleaned off.
I also like to grow the clean new parts separately to get plants that grow healthy in water culture for years. If the plant is a fast grower l don't even bother keeping the old sections that involve much more cleaning and retain the decayed parts which I don't want to have around. My Anthurium cristallinum has produced over a hundred offspring this way in the last 5 years. As you can see in this photo a shoot that developed 2 inches below the water level ( which is indicated by the red pen in the photo) has just emerged from the water. Some of them are coming from much deeper in the water.
If you have any questions I will try to answer them.