Honestly, if it's posted you would be trespassing even if it is designated trout stream. You need to look for the streams that have and easement on them, otherwise you need to get permission. In my experience sections 1, 2, 4, and 5 have been the best for ease of access to all the various streams and the fishing has generally been better for me there. Otherwise whitewater state park is always a popular destination, easy to get to, and is pretty good, but you really need to check the current trout maps and look at the signs next to the road. The whitewater has quite a few areas that have special barbless hook regulations, size limits, bait restrictions, etc. Different parts of the river have different regulations so do your homework.
Some streams worth mentioning:
Map1. Forestville Creek and Canfield Creek in Forestville State park, access is relatively easy and the fishing is usually pretty good, make sure to have a few parachute adams and blue wing olives with you at this time of year. Camp creek in Preston has been really productive so far this year but it's a catch and release only stream. While you're in Preston stop by and check out the National Trout Learning Center which just opened up this year. You may want to check their web site to make sure they are open while you’re down, http://www.nationaltroutlearningcenter.org
Map 2. Duchee Creek south (upstream) of the fish hatchery has been pretty good, easement nearly the whole length of the creek and has some nice size fish in it. Since it's upstream from the fish hatchery it does get fished a lot. Gribbren Creek is pretty popular and access is good, has some deep holes that will hold some big fish, but it does have a slot limit on it.
Map 4. Mill Creek in Chatfield has easy access, easements most of the way, and is stocked regularly with rainbows, the browns in that creek are wild trout and I've caught several over 18" - all north of town. It does get fished a lot though so keep that in mind. One of the hottest streams in SE is Trout Run, the name about says it all.
Map 5. I've never caught anything real big on Big Spring Creek but if you looking for action and size doesn't matter then Big Spring can be a fun stream to fish. Rush Creek has good numbers of fish as well and is stocked pretty regularly.
I'm sure others can give you some other spots to try but these are the Designated streams I have found to be pretty reliable. One thing to note, some of the best trout fishing I have found in SE MN has on streams that are not designated. Don't be afraid to stop at a bridge and take a look at a stream just because it's not a designated Trout stream, some of the trout stream tributaries hold big browns in good numbers, I would guess because they are not fished as often.
Hope this helps, good luck.