We purchased our Odyssey brand new and currently have a little over 100,000 km on it. The only concern we have had is problems with the sliding doors. The most common problem is sticky doors. This spring my 6 year old daughter was opening the driver side sliding door and it came off the track and nearly landed on her. The only part holding the door on was the arm at the bottom of the door. The dealership said it was the plastic roller that broke and blamed me for not having it lubricated properly. They did cover the cost of repairs after numerous calls to customer relations. It has been a couple of months since this happened and now the same door will not latch properly.
After researching other complains about the 2005 Odyssey, it seems there are many issues around the sliding doors. I find it difficult to believe that a company would rely on plastic components for a sliding door that would get so much work over the life of a vehicle.