Is slow cooking safe?
I suspect that you're not talking about 'slow cookers' but 'sous vide' as you mentioned 'new trend'.

Although food is often cooked at temperatures within the 'food danger zone' (below 140°F / 60°C), the increased time at that temperature results in pasteurization of the food item when done correctly.

The recommended temperatures to cook food so you can be assured that you've killed off bacteria and such are based on having it there for only a few seconds.  The further below that temperature that you are, the longer time that's required to kill off the bacteria ... although there is a lower limit of when you can't achieve pasteurization (practically it's 130°F / 55°C)

For more details, see http://www.douglasbaldwin.com/sous-vide.html#Safety