For Eight (Square, 48 bars)
1st man cast and weave the set, followed by each dancer he has weaved in turn. Each dancer starts by casting Left and behind the dancer on their Right. Then he/she dances into the center when it's next possible and out of the set through the next hole; etc to place. Try 2 bars per start and move fast, or 4 bars per start and move easily. (Syllabus of Irish Dances, Terry O'Neal)
For Eight (Square, 48 bars)
Based on workshopping from the March 28th, 2010 Ceili Picnic, this is the best way to perform this figure with 4 bars weaving.
(4) The 1st man casts over his left shoulder. This cast takes four bars.
(2) The 1st lady casts over her left shoulder. Her cast, and all subsequent casts, take two bars. While the 1st lady is casting, the first man weaves left into the set and forward.
(2) The 3rd man casts over his left shoulder. While the 3rd man is casting, the 1st lady weaves left into the set and forward, and the 1st man weaves right out of the set and forward.
And so on, for the next 40 bars. The 3rd man is followed by the 3rd lady, 2nd man, 2nd lady, 4th man, and 4th lady. As the final weaver, the 4th lady often has to cheat home on the final bar.
For Six (Triangle, Currently 40 bars)
The hey works the same way, but the figure ends up being 40 bars. With the eight remaining bars, you can swing, ring, or chain as desired.
Another option, which is as-of-yet untested, is to have each dancer cast in place over their left shoulder before beginning to weave, and upon returning home. This needs to be workshopped to ensure that the person casting in place is not a hazard for those still weaving.