I have been looking for the same thing (to remove the div that wraps around the content in the text widget). For example, I want to manually input HTML code of a list of things, so I have added ul before widget and closing ul after widget, now all I would want to do is in the text widget input the li’s but these li’s are being automatically wrapped with that div so it would be cool If i could just stop that div appearing completely.
I understand if I change the core file…I will have to do it again and again every time i will upgrade wordpress, so wondering is there any simple way of doing this and avoiding to re-change the code once i upgrade my wordpress?