See my co-founder's response, pasted here:

Neil is correct, spinach and lettuce grow best not in direct sun as well as cauliflower, broccoli, and arugula, they will do something that is called "bolting" when they are over heated. This means they will send up long stems with flowers, once they bolt they are no longer good to eat. Kale will do well with indirect light but they are more heat tolerant than the others I listed. Also melons such as watermelons can take direct morning sun, but require shade or indirect light during the peak heat of the day ~11 am to 2 pm is considered peak heat of the day during the summer months. Like neil mentioned coleus will do just fine with indirect light however the colors may not be as vibrant. As far as flowers go Hosta, Ivy, Iris, and geranium do just fine in shady areas. Black eyed-susans will also grow in areas that have more shade than sun :)

as far as herbs go basil and lemon balm will do just fine in partial shade, just make sure with all of your herbs that they are in a pot that has good drainage. Basil and rosemary will get something called root rot if they are not well drained and this can allow fungus and other bad diseases to thrive if they are over watered or not draining well. When growing plants for shadier gardens it is best to start them indoors, this will give them a jump start and bring them to harvest faster once you put them outside. As with all plants that you start indoors, including plants that require full sun, you must give them a hardening period. Hardening is when you put all of your seedlings that you started indoors in a full shade protected area outside or on a porch that is exposed to open air. Let them stay in this area for a few days then you can implement them where ever you like out doors. This is very important other wise they will be "shocked" by the elements.