Just like cleaning and chores, you have to take advantage of what your partner can do well. Guys are happy to cook as long as you don't make it seem like it's all on him to be the Iron Chef.
Always make it seem like a team effort. You may be asking him to turn on the stove and stir the pots that night, but you can help plan the meal. If he's shy around the kitchen, put together a few easy recipes and make sure all the ingredients are always on hand. Work together the first few times until you both know them by heart. If he's good with a few family recipes, make those part of your regular rotation and, by all means, never turn your nose up at a chance to barbecue. The stereotype is true; guys really can't resist an open flame.
Time to clean up