I carry the Fire stick in my bag with remote. It takes about 2 minutes to find the HDMI and USB (for power) ports and even the pain-in-the-ass Wi-Fi hotels throw at you.
I did require an HDMI connector swivel from BestBuy (my gen 1 stick came with one but I misplaced it), but otherwise all has been seamless. Keep in mind that your content is unlikely to stream for most titles as part of any package (Netflix, Amazon Prime or otherwise) when out of the country, but you can "buy" them and then have access.
Makes travel life just a little more like home.