I have always sung "Sunrise, Sunset" from Fiddler on the Roof. It is very easy to sing and has sweet lyrics.
The Wedding Song is a good one, too (I think that is what it is called): Don't know who wrote it, but it has nice lyrics also, though they are pretty traditional:
"well a man will leave his mother and a woman leave her home; they will travel on to where the two will be made one. As it was in the beginning is now until the end. Woman draws her life from man and gives it back again. And there is love. There is love. Well then what's to be the reason for becoming man and wife. Is it love that brings us here or love that brings us life. If loving is the answer, than whose the giving for. Can you believe in something that you've never seen before? Then there is love. There is love"
There are more verses, and I might have some of the words wrong, but that is the idea.
Also, Tis a Gift to Be Simpe is a very easy, very folky tune that almost everyone knows.
There is a beautiful one by Steven Curtis Chapman called "I will be Here". He is a Christian singer, and the song is a bit over done at weddings...but it is gorgeous! And I don't think it is too "god-y". I have the lyrics for it if you want me to post them. I also have the chords. Actually, I have the chords and lyrics to all these if you are interested.