First, if fish (or other marine animals) were eating the plastic (and there is a lot of evidence that they are) then they would also be starving to death (as they can't digest the plastic, and it fills up their digestive systems). When they die, the plastic would be returned to the ocean, and we would see it in our assays. So I don't think that the plastic getting eaten is the obvious solution.
Second, as anyone who has gotten sunburned while swimming knows, water doesn't block ultraviolet light very effectively, so plastic floating near the top of the water column would be exposed to a lot of UV. Plastic breaks down pretty quickly when exposed to UV, so we may just be seeing the natural destruction of the plastic by sunlight (and, I suppose, that plastic that has been partially broken down by exposure to UV might be more easily consumed by bacteria, but that's pure speculation).
Third, maybe we aren't measuring the amount of plastic in the ocean correctly. If the plastic is being consumed, or is sinking to the ocean floor, then we might easily be missing it. Also, the plastic might well not be evenly distributed across the ocean: it may be collecting in specific places due to winds and ocean currents. If we are not collecting samples evenly over the entire ocean, then we could be missing some high concentration areas.
I doubt that this means we can all breath a sigh of relief and decide that dumping plastic in the oceans is no big deal. I also doubt that this means that plastic is a much bigger problem than we thought (how could it be a bigger problem then we thought? People have been screaming about it like it was a sign of the end-times!). It is interesting, however, and I would like to know why our measurements don't match our expectations.