This may help isolate the problem. I assume you are using something like minicom on the Pi to test, configuring it for /dev/ttyAMA0 (hardware port). I have also used USB to serial adapters on the Pi and they work ok. When configuring minicom (sudo minicom -s), be sure to turn off hardware handshaking. Some reports elsewhere on this site report the ESP coming up at a weird baud rate after a firmware update.
Are you using 3.3 Volt power from the Pi system board? Or perhaps through a USB to serial adapter? This is probably not an issue but a separate supply to the ESP might be worth a try. The module should have around 250 mA available to it.