How important is visually indicating that ListBox items are drag-dropable in an application where not every List has this functionality?

We're building an application with various modules a great deal of which involves browses with lists of items (not unlike the list of questions here on Stack Exchange). One of the modules (thus far) involves a page/view with a few lists next to each other that allows you to drag items between said lists fairly similar to Trello (like so):

The core of my question is that Trello has no real visual indication that you can drag and drop these cards (possibly because it's a primary feature of the webapp itself?), so does there need to be some indication (e.g. skeuomorphic grip dots that appear on mouse-over for each item) that our own list items can be dragged and dropped?

Some small additional things to consider are that this is a line of business app that will be used on an all-day-every-day kind of basis so once users are aware list items are draggable in this scenario that's pretty much it.

Another is that we're considering extending the drag-drop UX to a few other areas of the application now we've got the actual code in a state it's fairly easy to attach to things, but this extension will appear on an 'as and when it's useful' basis.

Thanks for reading!