You don't put anything in them, particularly if they are more ditches than swales (less than three feet, top to trough). If you do put stuff in them, especially big things like logs, you defeat your own purpose of digging them in the first place, the better to retain water. Putting straw in the bottom does work for larger swales, which act as intermittent ponds, and hold water for long
enough periods for the straw to decompose and act as fertilizer; it also keeps weeds down. But you don't pack anything in by hand -- save that sort of work for zone 1.
About logs: their virtue is that they don't decompose fast, so use that virtue rather than fight it. Put them on the bottom of the mound,
hugelkultur style, burying them deeply enough so that they don't decompose fast, along with any big rocks. Otherwise, if you want them to rot, keep all stumps and logs covered with mulch, and wet, as Scott Nearing mentioned in 'Living the Good Life.' They will rot in their own good time, you need do no more, although throwing part of a rotting log on top of them will introduce the right fungus to them. A low spot would be good for that, but not a swale!
For even lower swales, plant vetiver grass on contour instead.
By the way, according to a Lawton
video on ponds and swales, you
should have a flat section at one end of the swale to allow overspill into a gravel spillway. It is a good idea to combine them with a
pond that they can
feed into.
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