What could cause flashing LED lights when other LED lights in a different room are turned on?
Some LEDs with switching power supplies demand power in pulses, which can get coupled back onto the power line as small surges. If you have bulbs which are making that demand at different frequencies, and the current available is a bit marginal, this can sometimes make one or more take longer to charge than they should and cause flashing.

I have a chandelier which takes 16 bulbs, which is something of a worst case since they are all trying to draw power through the same ten-foot  lamp cord and am using a dimmer to control its brightness. When ask the bulbs were identical, there was no problem. When one failed and I replaced it with a bulb that drew a tiny bit more power, some dimmer levels would cause that new bulb to flash badly. Replacing it with a bulb that drew slightly less power than the others solved the problem. I have some guesses about exactly why, but they are more detailed than I think we want to into here.

So I'd try making sure all the bulbs are identical and see if that helps.

Some folks have  reported that replacing one of the bulbs with an incandescent or fluorescent may also solve the problem, since these draw power differently. Again, I have some guesses about why, but they are mostly guesses. Might be worth trying an experiment, though.

We're playing with a product that's still in the early stages of its evolution. There are likely to be a few hiccups before the manufacturers learn how to make them "as cheap as possible, but not cheaper." I am trusting/hoping that those companies are aware of this effect and will solve it eventually. Unfortunately, given the long life of LED bulbs, that may not help us early purchasers very soon.