Broadleaf herbs, such as mint and parsley would do fine in your location. Narrow-leaved herbs, such as rosemary, need full sun to do well. Thick, narrow leaves just aren't as efficient at producing food for a plant as are broader leaves.
Greens like lettuce should do fine, but plants with darker green leaves might do okay, too, but would probably do better with at least 6 hours of sun.
Plants that produce fruit, such as tomatoes, really need a minimum of 6 hours of sunlight daily.
I think you should experiment a little and try different greens and herbs in your raised bed. You may find that you're quite quite satisfied with the performance of a particular type of plant, even if the results aren't "textbook perfect".
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams