How can I prevent these LEDs from blinking when turned off?
My guess is that you have a dimmer which has a light or LED on it.  If this is the case, your particular dimmer does not have a dedicated neutral on it, so in order to power the light/LED, it needs to feed a small amount of current through the lighting circuit.  If you had a standard incandescent bulb, this would not be anywhere near enough to cause it to light up, but with CFL's or LED's, this might be enough to cause it to flash.

Purchase a dimmer that has a neutral connection on it (usually these are a bit more expensive). Because it can be powered without powering the rest of the circuit, it will eliminate the flashing light.