Do LEDs care about too much voltage?
To answer the question you asked: Yes, assuming you count turning into a puff of smoke as caring then they care about getting too much voltage (or technically too much current). 

However you've made a couple of fundamental errors in your circuit analysis.

You have 240 V over 12k ohms plus 5 LEDs. As a first approximation LEDs can be considered a fixed voltage drop of their forward voltage. Assuming the LEDs forward voltage is 2 V that means 10 V over the LEDs which leaves 230 V over the 12k resistor. Using ohms law on the resistor gives a current of 19 mA.

So you have 2 V over each LED and 230 V over the resistor not an equal voltage over each device.

The other fundamental error you have is that 240 V AC is not 240 V peak, it's 240 V RMS. 240 V mains peaks at about 340 V (RMS * sqrt(2)). At 340 V you have a current of 27.5 mA (330/12k), probably not enough to overheat and blow up your LEDs assuming they are standard 20 mA max parts since the 20 mA is a sustained current limit rather than a peak current. However it is more than you were intending in your design.