The problem I see: You are putting LED's in series.. Now this might, or it might not work, if they are PERFECTLY matched it will work.. If not, depends on the degree of mis-match. Odds are fairly good you may enjoy success.
120vac RMS is 120 zero to peak 120Z*Squrt(2)/2 (About 0.707 Full wave rectified you get the same voltage... If you do a peak to peak (Voltage doubler) you get 120*Squrt(2) (about 1.414
120 *.707 = 92.4,,, Three of those LED's in series is 99.. That looks like a good match to me.
NOTE unless there are current limiters in the LED assemblies (in which case exact matching is more important) varying the voltage will vary the brightness over a fairly decent range.. Making them too bright, however, will shorten their life.... Big time.