It might help suggestions if you were to list your equipment. What brand modem and router?
You said your other devices work with your internet...what kind of devices? Just trying to get a sense of what your actual connections are.
I have cox cable and have no problem with my wireless connections. I use a Linksys modem that Cox approves of, a Linksys router, have my Roku and 2 PCs connected wirelessly. Of course I also have TVs and VCRs connected to the cable analog signal. In my router, I set up a list of MAC addresses the router should respond to, and of course, Roku is one of them. Roku has to ping the router quite often as it checks to make sure it is still connected to the Internet.
I think if you have other devices and you have had no problem accessing the internet using whatever devices need to do that, then you should be able to have the Roku access it also.
The trick is always to find out what is causing a problem. So the more information you can put here, the better help the members can be.
Trying the above posted suggestions you have received so far, is a step in the process of eliminating what doesn't work until what does can be found. Do you subscribe to the Instant streaming by Netflix?
Original Roku 1, called N1000,bought 2008 (2014 still works perfectly)
2014 new owner of Roku 3500 (HDMI) streaming stick