In reply to KA:
The main multi-pitch climbing areas are, from west to east, Rätikon, Tannheimer Tal, Wetterstein, Karwendel, Wilder Kaiser, Berchtesgadener Alpen (lots of areas around here, like for example Loferer Steinberge), Hochkönig... and it goes on further after that, but that's the furthest east which I have been climbing.
I am not sure what you mean by easy access. You have got to walk for pretty much all of them. The Stripsenjochhaus in the Wilder Kaiser does have many climbs which start very close by, but the hut itself is an hour and a half's walk from the car park. The climbs on the Wetterstein south side, for example, (Schüsselkarspitze / Scharnitzspitze) are about three hours' walk from the valley, or one and a half from the hut (Wangalm), while the most worthwhile climbs on the Wetterstein north side are those reachable from the Oberreintal Hut, itself about a four hour walk from the valley. There are some valley-level crags with multi-pitch routes on them, for example the Martinswand near Innsbruck. However, I wouldn't even drive the sixty miles from my house to climb there, much less come all the way from Britain to do so.
I see from your profile that you climb E3, which means that you will have lots to go for here in the range of UIAA VI and VII., and these are the grades you need to be climbing to get the most from the multi-pitch routes here. My suggestion would be to take a few days climbing the classics on the Schüsselkarspitze south face (e.g. Rainer / Aschenbrenner, Peters / Haringer, Bayerischer Traum), and then a few days at the Stripsenjochhaus climbing the Wilder Kaiser classics, e.g. Dülfer on the Totenkirchl and the many routes on the east face of the Fleischbank.
One area which I don't know much about but which looks great if you are climbing E2 and above is the Hochkönig massif. Start by looking into the routes Gloria Patri and Dientener Weg on the Hochkönig itself. In total there is almost a lifetime's worth of climbing in that area alone, and easily enough to fill one whole trip just picking off the best known classics.
To finish, not exactly a tip, but more of a challenge: The Rebitsch / Spiegel and the Rebitsch / Lorenz on the Laliderer north face in the Karwendel. First climbed in the 1940s, no UKC ticks, and I don't know anyone here who has done them either. Apparently not routes for the married man.
