Not the organic foods.
Not that broad a definition, either species-alien genes have been inserted or they haven't.
The issue with them is that we've gone GMO so fast that we have no clue what the long term effect will be, and some harms have already been identified. For example, one type of GMO makes crops more resistant to pesticides so that more pesticides can be dumped on them. In the case of "Round Up Ready" GMO crops, the consequent loss of milkweed is wiping out the Monarch butterflies.
What if people were more like bees?
Many people are quite concerned with personal health. I'm more worried about environmental impact, including the spread of GMO genes to natural plants and animals. Organic farmers have even been sued for patent violation because their crops were contaminated without their consent by windblown pollen from neighboring GMO crops.

:cussing:
I'm no expert, though. There's lots more at the "GMO Labeling 'Compromise' Is a Cop-Out" link above.
The specific issues here are that consumers should have the right to complete information so that they can then make better choices, and that the Big Brother feds have now stopped that from happening in VT and elsewhere at the behest of Big Ag. If you want to buy products that are accurately labeled GMO, no one is trying to stop you.