Just like newspaper and junk mail, cardboard items can be throw in the bin as a dry material. Remember to shred the cardboard into little pieces before composting. Common cardboard items around the house include toilet paper rolls, paper cartons from eggs and berries. Just keep in mind that all cardboard should be clean and grease-free. So, can you toss in that pizza box? Yes, but you should remove the oil-soaked portions first.