Can I paint my kitchen cabinets if there are three different textures?

Yes you can but all three will look different once you apply the paint. You will see more of the texture in the particle board, you will see the grain in the solid wood, and the laminate will look smooth.

Try chalk paint. Be sure to clean really well with TSP and to read the directions for the paint. It dries really quickly and may need thinning with water as you go. Just finishing our kitchen cabinets (two coats and a watered down third coat and then waxing) and they are looking great. You can also use it on countertops and then coat with a sealer. We decided to go with the Giani countertop finish in White Diamond. I'll let you know how it turns out.