I've recently been on a Carthy workshop. His tuning is CGCDGA. Alistair Hulett uses DADEAB. He got that off Martin Carthy (Up one tone).
I've been concentrating on CGCGCD, learnt from Gavin Davenport (of Crucible and Hekety) at one of the Hands on Music Weekends, Strings at Witney
The nice thing about this tuning is that traditional songs and tunes seem to fall right under the fingers and it is difficult to play a wrong note - it is usually either right or a harmony! The downside is that you need to use a capo lots (to play in D, capo at 2nd fret, to play in G, capo at 7th fret). This tuning also conforms to the Martin Simpson method (see Weelittledrummer above) where he explains that his tunings are just root, fifth and another, the root being C, the fifth being G and another being the D (either 2nd or ninth) . Open C would be CGCGCE (r5r5r3), Open C minor would be CGCGCEb (r5r5r5flattened3). DADGAD is r5r45r. Open D is r5r53r and open G is 5r5r35
On another Strings at Witney course led by Chris Foster, we learnt EADEAE and its variants DADEAE and DADEAD (r5r2nd5r), which is not much different to the Carthy tuning (although a tone higher).