VOLTAGE DROP IN WIFI MODULE
The ESP8266 current requiremnts are as follows: 


If your Lm317 is working correctly then it should be able to supply this current without any problems.
You need to ensure your input voltage is at least 6.3 V (minimum Vin - Vout for the LM317 is 3 V). (There is also a minimum requirement of 10 mA so you won't be able to achieve low current such as when the ESP8266 and the driving microprocessor are in sleep modes)

You should be able to test your regulator with a 15 Ohm resistor. If this works you likely have a bad ESP8266 ....if the voltage drops below 3.3 V you have a problem with your regulator.