If you've tried downloading anything from Amazon Cloud Drive that isn't an MP3, onto your Android device, you've probably been pretty annoyed at the experience. On the default browser (and other shells around it like Dolphin) you can't select individual files, and the list scrolls when you attempt to click (well, touch) individual links.
Neither Opera Mini nor Opera Mobile work either - you get a bizarre page that tells you you need Flash to upload files, that doesn't let you navigate from it. And you get it even if you've enabled Flash.
So... what's the solution? Firefox. ACD seems to work fine with Firefox for Android. Just be aware that Firefox doesn't work under half the Android phones out there because it's been compiled for a specific CPU architecture, but if you have a phone or tablet capable of running Firefox, you can use that.
(If you use Firefox as your desktop browser, you'll be interested to know it syncs very nicely with the mobile version. A neat trick is that you can load a page you have on your desktop onto your tablet just by selecting the tab from your tablet. Very nice for that "Hey honey, take a look at what I saw on the web" moments.)
If you're using a Kindle Fire, you can download Firefox directly from this link.