Same problem here with Chrome 15.0.874.83 beta & Ubuntu 11.04. With Opera 11.51, it just shows the Markdown source. With Konqueror 4.6.5 it opens the file in gedit. I suppose these are all different symptoms of the same issue which is that the drop event doesn't get handled as expected.
It works great with Firefox. Very interesting tool!
You are using it correctly, so must be missing some support. You might mention what version of Chrome (although, I assume that's the latest given their automatic updates) you are using and whether or not you are blocking JavaScript.
Using Ubuntu & Chrome.