Running an Arduino off an AC Circuit with a Bridge Rectifier
Your diagram shows an LM340MP-05, which is a 5.0V regulator. I'm going to assume this is a mistake in your diagram rather than that you picked the wrong regulator, as you mentioned you'd measured the 3.3V output and it was ok.

A few thoughts I have on what might be wrong:


You're connecting the 3.3V output from your regulator into the VIN pin on the Arduino, rather than directly to the 3.3V pin. The VIN pin on Arduino boards usually goes through a 3.3V regulator to drop your input voltage (usually 5-9V) down. The regulator will not operate properly without a bit of voltage overhead (called the 'dropout voltage') above the target voltage - giving a 3.3V linear regulator a 3.3V input will probably result in about 2.9V at the output for an LDO, which would mean you're undervolting your Arduino.
You're cooking your regulator with too much current at the given voltage drop. The LM340 offers 1.5A maximum, but linear regulators burn power proportionate to both the circuit current and the voltage drop. You're dropping 22.7V and an Arduino's idle current draw is about 50mA - this means you're burning 22.7 &times;  0.05 = 1.135W of power in the regulator just by running the Arduino at idle. This will produce a temperature rise of about 27C if you're using a TO-220 package regulator, but on the SOT-223 it'll be more like 70C above ambient, which is enough to start causing problems. Keep in mind that these numbers assume you're pulling only 50mA; any more and the heat problems will be even worse.
You've placed a 100uF capacitor for smoothing the input, but you haven't placed either of the input or output filter capacitors for the regulator as recommended in the datasheet. This is unlikely to be the sole cause of your problems, but a 0.22uF capacitor across the regulator input is good practice.