Is there a straightforward way to turn LED lamps on and off using X10 devices.
If the answer is no, then is there any system which can perform the above task.
Well the best simple answer to your first question is: NO.
For a more qualified answer a lot depends on:
1. If your wall switches have neutral lines.
2. If you use dimmable bulbs
3. If you desire to dim the lights.
If your wall switches do not have neutrals lines, IMHO, NO there is not a guaranteed straightforward way to control an LED without flickering or erratic behavior sans an incandescent bulb in the string.
If you use two wire switches because you do not have neutrals in your wall boxes then use dimmable bulbs, and CREE are good. Remarkably I have good non flicker service from 100W equivalent Sunbeam bulbs from Dollar Tree. Higher wattage (equivalent) dimmable bulbs have higher levels of leakage current and two wire switches like PLW01 may work acceptably (this is really case by case depending on bulb brand and wattage).
As far as a system that does work, you might look at Z-wave. You could try asking the same questions on the Homeseer forum. Homeseer can control X10 or Z-wave so there are many Z-Wave users on that forum which can tell you more. Z-Wave is an open standard with dozens of brands most of which are about three to four times more costly than X10. But I think two keys for stable operation is: a neutral in the switch box (and use a switch that requires a neutral), and using dimmable bulbs whether dimming or not.