Replacing incandescent bulbs with LEDs, unexpected behavior
Perhaps the dimmer circuit won't work without a resistive load. Some thyristor dimmer circuits might not see enough holding current from a rectifier-capacitor load to stay on for the half cycle. You may be able to wire a fat resistor in parallel with the LED lamps and have it work, however it may require some experimentation to determine an appropriate value.

Here's a typical small triac datasheet. The holding current is specified as maximum 10mA or 20mA, depending on type. If a phase-control trigger circuit provides a short pulse, the MT2 current must reach the holding current during the trigger pulse or the triac will not stay on for the remainder of the cycle. (bottom line below). 



Here (from Wikipedia) is what the output of a typical phase control looks like. The gate trigger pulse would take where the vertical red chevrons are located. (a) is the power waveform. (b) is the output with triggering near the beginning of the cycle (bright) and (c) is the output with triggering near the end of the cycle (dim). 



I would try about a 150 ohm 2W resistor -- if you get a few you can parallel several and go up in ~1W increments. You may find that too high a resistance causes half-wave triggering (intermediate brightness), since triacs have different holding currents in the positive and negative directions.