One problem related to radiator has been reported for the 2004 Honda Odyssey. The most recently reported issues are listed below. Please also check out the statistics and reliability analysis of the 2004 Honda Odyssey based on all problems reported for the 2004 Odyssey.
Auto repair shop informed me that two motor mounts were broken. Not repairing them could result in the engine dropping and pulling radiator down which would be dangerous and result in a more expensive repair. They informed me that motor mounts are not a maintenance item, like belts, for example. They should not have to be replaced in normal circumstances. They also said they have seen a lot of this exact problem in this generation Honda Odyssey. Further, the price of these motor mounts is very high and has not come down which is what they would expect given the frequency of repair and age of model. I called Honda America which told me that since my car is out of warranty there's nothing they can do.