I would try to create a lush jungle feel as opposed to anything mediterranean. Think lush jungle, think serene woodland, think foliage, think large leaf, think elegant ferns, think climbers, think evergreens, think water features, think stone, think garden art and garden furniture that can be just as pleasing as sun loving flowers. Think mirrors for capturing light. Your garden might not see much sun, but that means the soil wont bake hard in mid summer, so you can grow shade plants that will thrive in rich damp soil.
In areas that do perhaps get summer sun, grow those sun worshippers that you cannot live without. Some plants can survive on as little as four hours, if its good strong sunlight. It will be trial and error of course, personally I'd get rid of the lawn (grass needs sunlight to grow nicely) and indulge in a forest of foliage and climbers. You'll be surprised how many gorgeous plants grow in shade.
even certain cultivars of roses will grow in part shade. Monty did a feature on GW last year. 