To export all attachmets from a trello card:

Open the Card and Click "Share and More" in the bottom right corner
Click "Export JSON" in the resulting menu
Copy all text that appears on the screen and paste in the box below

Update 13 Sept 2021 - This will no longer work for almost all boards.
Trello have added authentication to download URLs which means they cannot be accessed to consolidated from this tool.

I hope this was useful while it lasted... RIP Trello Attachment Downloader

This also works for an entire board, just click the "Show Menu" on top right of board.
Then click "More", "Print and Export" and "Export JSON" and copy the resulting text into the box above.

People have used this tool 66649 times downloading 3 TB and 2081009 individual files

This software is provided in the hope that it will be useful, but without any warranty;
without even the implied warranty of merchantability or fitness for a particular purpose.