Well, unfortunately I have to report I had my first negative experience with my concierge. I don't want to get into specifics, but I asked my concierge to contact a property in order to resolve an issue preventing a reservation via the website. She replied claiming that I misidentified the issue, but it was obvious she never contacted the hotel at all because when I called the hotel directly the next day, they were able to get me set no problem - and in about 5 minutes, at that. Unfortunately, I'd wanted to reserve a block of rooms at that property and in the time that passed, the hotel had sold out of the room types I needed.
While it wound up being a minor issue, it represents a bigger problem because a concierge should never leave things worse than not having the concierge at all.
This experience and a lot of the dissatisfaction I've read in this thread makes me wonder if the "My Hyatt Concierge" wouldn't be better as simply a centralized e-mail center of well-trained, better-motivated agents. I get the idea of having personalized service from a one-on-one relationship with a single concierge, but there seems to be so much variance between agents and even in experiences with the same agent.
And don't get me wrong, I still think it's a very beneficial program in its current state - even just for resolving administrative or billing issues, it's saved me tons of time by being able to e-mail one contact and get things accomplished. I just wonder if it could be improved.