Get rid of JIRA, do not have any tracker at all, communicate things verbally only? I have seen this work very badly... things dropped on the floor, forgotten features for release. And it makes inter-team collaboration much harder: if you have a request for other team, you have to either attend their meetings, or ping them in person with "how is the progress" questions... which makes it easier to just ignore other team and do the thing yourself.
Get rid of JIRA, replace with other "Agile-oriented" ticketing system? I haven't seen those in action, but that would be an interesting blog post to read.
Get rid of JIRA, replace with other simple ticketing system like TRAC, Trello, or Github Issues? Meh.. You can duplicate parts of JIRA out of anything -- even in github issues, you can make a label for "sprint" and define "epics" using tickets which reference each other. Or you can prohibit regular people from creating tickets.
Get rid of JIRA, replace with ad-hoc things like email inbox, google doc, or a wiki page? I think this works when there are only a few people, but once you have more than half a dozen you really need a good discipline to avoid making a huge mess.
Get rid of default JIRA setup that some Agile coaches talk about? That would make much more sense to me. Wouldn't make a very catchy blog post title though.