Have you taken a look at my 'fixing a muddy run' page, link in .sig below - a multi-pronged approach usually works best.
As far as your exact question, an open trench or swale actually works as well or better in many (not all) climates/soils. If you peel the turf back, then excavate, then replace the turf, it doesnt' take too long for the grass to knit back together so you have a grassed swale rather than a bare mud trench (although it *is* a lot more work that way).
Rather than give specific dimensions I think the most useful thing to say is "build it as big as you can stand and seems reasonable given your soil type and rainfall amounts".
A little trench 1 shovel wide and deep will actually do a surprisingly decent job if it is graded correctly so the water is all led *away*; it will fill in fairly quickly if open, but if lined with (ideally) 4" perf drainage pipe in a 'sock' and covered with some gravel and turf, and the end of the drainage pipe led away to somewhere distinctly downhill where the water can flow away freely, that'll be enough for most small coops.
If you don't want to use perf pipe, dig a larger trench (maybe 2-3 shovels wide and 1.5 deep, but it depends a lot on your situation), line it with straw or landscape fabric, chuck in a bunch of largeish coarse gravel, then top with straw or landscape fabric and several inches of soil and your turf.
The above items only work OUTSIDE the run, btw -- if you try to do a french drain IN the run, the chickens will jsut dig it up and make a mess of it
But, don't neglect other strategies as well.
Good luck, have fun,
Pat
