In reply to Martin Haworth: You can also go for "Alpinism" in lower ranges that are normally free from snow in the Summer, e.g. the Belledonne massif just north of the Ecrins. At that time of year you can get alpine-like routes there, which are much harder/impossible to do any other time.
I do recall climbing the North couloir of Les Bans in June, not normally possible much later, so that goes with your comments about Ecrins couloirs.
May can provide superb weather and very empty mountains, last year gave a week of almost uninterupted sunshine. The sheer abscence of people gives anything above the snow line a quite serious feel though, more like when the Alps were terra incognita, while as someone mentioned, the huts will almost all be shut.