The closest legit trails are going to be Kansas rocks, down by Mapleton Kansas, and Turtle Creek Reservoir, north of Manhattan Kansas. Both of those are more technical, with rocks, and such. If you just want trails, and mud, there's some trails around Tiffany Springs Athletic Park. Keep in mind, if you do those technically you're trespassing. There's also Kickapoo Trail. It's nothing official, just two series of roads between Leavenworth Kansas, and St Joseph Missouri.
On a serious note . . . me thinks if you take 435 on the East side, get off on Garfield, head West just a bit you'll see a gravel road heading off up over a hill on the left. Have no idea where it goes, if it goes anywhere but it does look somewhat promising.
On the West side of town, again off 435 on Shwnee Mission Parkway head West within ½ mile or so. Somewhere in there I've seen a dirt road that disappears into the woods. Same details as above.