I repurpose the old laptops that I seem to accumulate for this. Just about anything made in the last 10 years can handle at least 1080p video, and will have an HDMI output. I add a small wireless keyboard and use my regular desktop environment. But you can add a remote and use one of the open source media centre suites.
I do wish more TVs had a monitor-like sleep mode and accepted the command for it over HDMI. And were actually low power when they sleep. That seems to be pretty spotty.