In reply to gareth rees:
Rather than giving you specific advice, I'd suggest you take a look at the Alta Via routes, of which there are many. Most of these are over a week long, but it wouldn't be very difficult to choose a section of one of them. If you google Alta Via Dolomites, I am sure there will be plenty of information. Personally, my favourite areas are in the area between Lagozuoi and Civetta (tyhis is incorporated in one of the AV's, I think 1 or 2 but can't remember which, with the section from Nuvolau to Civetta being particularly spectacular. It's quite moderate walking with plenty of over night stops on the way. I'm guessing that particualar section would take 2-3 days depending on how quick you walk? Sella pass is also stunning, but it's so damned crowded in summer these days that it can feel a bit like a circus. Although mid June is a little bit earlier than the main season so you should have a good time. Make sure you book ahead - I would be asking now because the hut get pretty booked up.