Interesting thet the D4 marketing literature say "Through dovetails
1/8" to 1-1/4" thick" with no mention of a limitation.

Reading the D4 documentation it actually shows a limit for the tail
board as 1" with info for up to 1 1/4" for pins [Table 15-6 page 134].
To go thicker than 1" on the tails you will have to use some method
other than the Leigh D4. Here are some ideas. I think if it were at all
possible to do it with the jig they would tell you how.

1. Use the D4 to cut the pins then use them to layout the tails and cut
them by hand.
2. Do the whole thing by hand.
3. You could do them with 5/4 on the long sides and 4/4 on the short
sides if you are building rectangular boxes or even if you aren't.
Symetrical is more common but not a requirement. You could even mix it
up more and build some smaller boxes too, with thinner pieces to add
some character to the whole set of boxes.
4. You could get real creative and have all 4 sides step down, bevel
down or taper down on one side or symetricially to 1" in the last few
inches at each end. If they had a square 1/4" step on the outside, this
would provide some nice locations to kind of lock in the boxes as they
stack in certain configurations.

Are you sure your material will still be greater than 1" after planing
and smoothing?