As of October 2015, it has changed a bit, and I decided to post a new answer, as this is the first Google result!
According to a forum post, the only way to remove attachments is to open the problem in the Problem view and not the Projects view, click on the problem, scroll down to the attachments and then click on the trash can icon when scrolling through the image.
Source: Jira Forum
This is not possible directly in the detailed view of the green bush. First you need to open the problem in the normal JIRA verbose view by clicking the Problem button in the greenhopper verbose view. In the regular JIRA view, you should see the trash can icon if you hover over the attachment. If not, you should check your permissions.
D. Gibbs
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