The dirt that you are seeing is probably just marks made by rubber conveyer belts that are very common in the shipping industry. That's not a problem at all for the recycler. That all comes out in the wash. By the time it gets made into new packaging material it's usually pretty clean. If the final product isn't clean, it won't be used for food. If it's enough dirt to change the weight of the cardboard, then it's a lot bigger concern, and you should consider composting it instead.
If it looks like it might be grease or oil, then it's less useful for high quality paper products. You can tell it's oil if it has absorbed into the carboard and spread out in somewhat round patterns, much like how grease from your pizza soaks into the pizza box. You can identify heavy automotive grease if it spreads to another surface wiped against it. Grease and oil can also be identified by how a drop of water is prevented from absorbing into the contaminated area. It's usually recommended not to put that into the recycle bin, even though it might be recycled for other uses.
What's more of a problem is when there is a lot of tape and plastic labels, so you can help by removing most of that. Either way, the cardboard is probably not going to be rejected. To be certain, you should contact the organization that runs your recycle program and ask them. They usually make a good effort to be clear on what they can and cannot use, and the answer is usually just a phone call away.