Not sure what you mean by "contemporary", but one of my wife and my favorites is "My Cup Runneth Over" (or also titled "Sometimes"). This was covered by many artists but our favorite version is by Ed Ames. It is a very slow waltz. Here is a link to a shortened dance to this song by World Champions Michael Meade and his wife Tony: http://www.bing.com/videos/search?q=Michael+Mead&view=detail&mid=618F1E3A9D201415A033618F1E3A9D201415A033&first=0
Everyone wants something different in their music but for a wedding waltz, Come Away With Me by Nora Jones seems to fit what you are looking for; it is contempoary, romantic and meaningful (at least to me).
While you may be looking for something a little newer, listen to the Andy Williams version of "Charade." We lost Andy Williams recently, and all the obituaries seemed to say that he was the one who sang "Moon River." He did, and very well--but so did several other performers. But for my money nobody beats his "Charade." And the tempo is ideal for dancing.