[megatron]

First of all, you need to make sure that this is indeed a factory intact immobilizer and not some 3rd party system. I mention this because you said there is no blue light on the dash telling you the key is wrong... There is never a blue light on an 2003 Odyssey. There is a green key symbol in the dash that flashes when the immobilizer is not sensing the correct key. If it senses the correct key, it will light up for just a moment and then turn off when you have ignition on.

The way the system works is that the immobilizer unit has to be programmed to the ECU. And then keys have to be programmed to the ECU also. So if you replaced the immobilizer unit, you must first reprogram the ECU to accept and recognize that new immobilizer. After that, you have to then program the keys to the new immobilizer. All of this is done using the Honda HDS service tool. You cannot do this yourself unless you have access to the HDS system as well as to Honda's servers to get the immobilizer reprogram code that changes daily. Any Honda dealer should be able to do this for you, as long as it's a factory system and not a 3rd party one.