Go to JIRA issue navigator and add some criteria (or write JQL query)
Do search to see search request result.
Select Attachments from Export dropdown menu.
This will take you to Download Attachments for JIRA dialog.
Dialog allows you to:
Filter attachments (include or exclude) based on predefined extensions or you write custom wild card match expression.
Group attachments in folders based on selected grouping like: issue key, issue summary, extension etc.
Click Download. All attachments will be packed and served from browser.
Only most modern browsers will work with plugin. Total size of result zip file is limited by client browser and available free memory on clients computer. Details about performance limitations can be found here.