Boy, the theories people come up with! The flashing Tx/Rx lights is the ELM327 doing a "lamp test", it's not an "OBD handshake".
However, they are probably correct in thinking that the device is not receiving power (either that, or it is defective). If you have a voltmeter, measure the voltage between pins 5 & 16 of the diagnostic connector, it should be roughly equal to the battery voltage. This picture shows how the DLC pins are arranged, there are two rows and you count from left to right: