I have a 40g breeder, lightly planted. I had the same problem. I removed my plants and rinsed them off. Then moved them to another tank, a 20g long. The 40g, I killed the lights. Just ran ambient from the room, which is quite dim. I had this set up for a few months (had too much life going on and just left it alone). When I put the plants back in, my light was broken, so I switched to a used led light, a Current freshwater. It seems to doing fine now.
I honestly have doubts about if removing the plants and going dark actually killed off the cyano. I'm leaning more towards the different light spectrum.
I had cyano in my reef tank when I first started out. I ran a cheap t12 set up. I tried heavy flow, feeding less and manually removing it. As soon as I put on a decent light, it all went away and never came back on that tank.
My suggestion would be, change your bulbs.