At one point, we pointed out a new information system that we were thinking of putting into place to monitor and control the flow of certain inventory. Mr. Yokoi's wise response, suggesting otherwise, was:
"When you put problem in computer, box hide answer. Problem must be visible!"
They warned us in Scrum training that while JIRA, Rally, and other agile management tools look tempting, they hide work (especially muda) and make it harder to limit the number of things you're doing.
That matches my experience: My first few sprints, I made huge post-its to hold our user stories, and put them up on a whiteboard. It was annoying and tedious to do and keep synced with our ticket-tracking software (would Agile purists just discard the tracking software altogether?), but it did make it easier for everybody on the team to understand exactly what we were trying to get done that sprint.
There's another advantage to physical story cards: it forces you to clarify what exactly you're trying to get done. With a JIRA ticket, details can live in custom fields, attachments, linked tickets, etc., and it can be hard to figure out from the ticket what exactly it's about? What constitutes done? With a paper card, you've got room for a few bullets and that's it. You can reference details/assets found elsewhere, but anybody should be able to pick up that card and understand what you're trying to do.
Maybe it's time to go back to the post-its.