It sounds like you already have a TinyCam server running somewhere and you would like to view those cameras on your Roku device.
I don’t know anything about the Roku, but I would guess that a web browser is available for it. Open the browser and use the web address provided by the TinyCam app and you should be able to view your cameras on the Roku.
There are lots of cheap, Android-based TV boxes. Try Amazon. Make sure the TV box has Android’s “Play Store” else you won’t be able to install tinyCam. Most of them don’t have Play Store.
Add your cameras to tinyCam pro and activate its web server. You’ll need to forward ports and use DynDNS, OpenDNS or something similar since it’s going to be behind a router. Then you can use any browser to view your camera feeds.
Not sure if you can install browser add-ons. If you can, a cheaper alternative is to flash the RTSP firmware, forward ports and use your browser with an RTSP add-on.
Note that tinyCam’s web server is CPU-intensive. The TV box can get really warm. Choose a model with as much CPU/GPU oomph as you can afford.