I also strongly disagree with the suggestion that backpacking requires boots.
I almost always use Inov8 Flyrocs. I have used Terrocs and Teva X-1 (and tried several other trail running shoes). The Flyrocs have a great tread. When I'm hiking in wet conditions, I just plow ahead. If the water is getting sufficiently cold, I will wear neoprene socks. I use one pair of very thin socks (about 1oz per pair). I take an clean pair for each night.
Goretex keeps my foot wet. The Flyroc allows the moisture to be pushed out of the shoe as I hike. Quite some time ago I posted on this site tests of various shoes for loosing water weight when hiking under controlled condtions.
Hiking in December through February or in Winter conditions, I don't use the above solution unless I expect rather dry conditions. I do use Goretex trail running shoes in the winter, when I'm snowshoeing.
The Teva X-1 is great if you are in constant water and it isn't too rocky. For example, in the Paria they are great. For the Zion Narrows (rockey), the Flyroc is better. I don't like the thin soles of the X-1 in rocky streambeds.