You may get some surface rust on the cooking grates when they are subjected to the slow-drying process of a dishwasher, especially as they begin to age. I would remove them immediately after the rinse cycle and dry them by hand.
I would also pay attention to whether dishwashing machine detergent, which is harsher than a dishwashing liquid you'd use in the kitchen sink, has any detrimental effect on the finish of the grates. This isn't too big a risk, since both grates can be replaced for about $20 total (although the smaller bottom one must be ordered from Weber directly).
If you decide to do this, please let us know how things turn out so we can all benefit from your experience. I would especially be interested to know how the grates hold up over the long run using such a cleaning method.
Regards,
Chris