[Al Skierkiewicz]

Chief,
Jason is close. The voltage you are reading is an approximation of RMS and for 60 Hz sine wave it is pretty good. To convert to the peak you multiply the RMS reading by 1.414 to get the peak and then by 2 to get the peak to peak or 18.32 volts. This is the full voltage swing available across the output terminals of the transformer. Calculating for 40 ma of current through the diodes, 18.32-3.7(diode forward drop)=14.682/.040 amps=367 ohms. The next closest standard value is either 390 or 270. I would go with the 390.
Without a full wave rectifier, the diode is only going to conduct for half the cycle of the sine wave giving you half the average current at a 60 Hz repetition rate. You can try the 270 and see if it makes it any brighter. You didn't give any current specs on the diode you were using so I guessed it to be 40 ma. If your diode is different than recalculate the series resistance. At your current resistance the diode is being forced to sink about 150 ma or more than 0.5 watts.