In ShadyRaider's final circuit, the antiparallel diode/LED are in series with a resistor - there is no capacitor. That's why OperaHouse suggested the diode be in series instead.
I also took VE1BLL's suggestion and bought a cheap night light. It turns itself off when there's light, so the...
I WONDERED why the diode wasn't in series. I thought the way it was would let a low voltage small signal diode be used instead of a high voltage one, but then I looked at current prices and a 1N4007 is only $0.03 so that couldn't be it. As a kid (1960s) I was careful to not use too much diode...
Of course it flickers. You don't see it while it's sitting still but wave it around a little and you'll clearly see the led flash. It's off 60 times a second for more than 2/3 the time.
I hope you used a big enough resister - probably a 1 watt minimum. I calc your circuit is about 5 mA, so the...