We've had a rough, shaky time using Jira, but it's almost entirely due to internal politics and constantly-shifting business priorities. (aka CEOs and owners who are just megaphones for their corporate customers) We intended to replace our invented-here ticketing system (which has Zendesk integration) with Jira, but instead we integrated Jira into that invented-here ticket system.
On the other side, my team has been bit by a bug from 2008 which prevents us from changing any ticket's state to Resolved. We can Resolve tickets, but their Resolution remains Unresolved. Since the Resolved state is central to most of Jira's built-in features, this means Jira has been broken for me since we installed it 6 months ago. The Your Issues default display shows all my tickets, including my closed tickets. The interface is not customizable enough to build workarounds for this problem.
Also, the name (bastardizing the Japanese pronunciation of Godzilla) always feels somewhat racist to me.