How to dynamically change the currency used in a report summary?

I don't think it's possible. You can dynamically change the filter parameters (see http://www.shellblack.com/salesforce/deploy/custom-report-links/) or mimic clicking of [Show Details] by appending "?details=yes" or "details=no" to the report URL.

There's matching hidden input field but passing "&currency=USD" doesn't seem to give any results...

<input type="hidden"  id="details" name="details" value="no" />
<input type="hidden"  id="currency" name="currency" value="EUR" />

<input type="hidden" name="delrep" /><!-- yes, we can delete reports by visiting "?delrep=1" but we can't change the currency ;) -->

Perhaps it wasn't possible at the time this question was asked, but in the meantime it is possible.
You can remove the currency you set in the Currencies Using picklist in the Classic report builder through the metadata.

You need to download the xml of the report using your preferred tool. Then simply delete the <currency>EUR</currency> element from the xml, and upload the xml back to the server using the Metadata API.
This will remove the default EUR currency from the report and instead will use the currency set in the user settings.