The people who used to live there may not have minded when people walked through like that. If I were you, I'd go around the neighborhood and introduce yourself, say hey and all and then say that you don't know if they have kids or not, but you've seen several kids in your yard and lots of trash has been dropped there. Tell them that you would appreciate it if they would (if they have kids) make sure their kids know that "the new people" don't want anyone walking through their yard because you are trying to grow grass, (if they don't have kids) help you spread the word that you'd appreciate it if folks wouldn't walk through there because you are trying to grow grass. Lots of times the parents don't know that the kids are doing that, so it really does the parent a favor if someone brings it to their attention nicely. That's happened to me before, right after we moved to this town.
After a neighbor came and told me about my kids (and NOT in a nice way either) I told them to stay out of yards and if they needed to cut through one because it was the only way without walking a half mile or something to go around, that they should knock on the door and ask. If they get permission then they should be careful, avoid flowers and other plants, push bikes not ride them, and never drop trash. Also, pick up any trash they see in the yard and throw it away. We never have problems with them doing that now.
In a similar vein, my mother lives across the main highway from me. It's not far at all and I can walk there in five minutes. I walk down to the end of my street, cross the main road, and there is a house with a large property between the main road and the back street that she lives down. That property extends about a block and a half to my left and ends in a point where the streets connect. Nothing is planted there except some scrubby grass. It's full of trees and the people don't even bother to rake their leaves. When my mother first moved over there, I went with my kids to the door of the house and talked to the man who lived there. I told him that my mother just moved to the old people apartments behind him and we live right there (I pointed at our house, which he could see) and asked if it would be ok if me or my kids cut across when we went to see her. He said as long as we don't throw garbage, it's fine. I thanked him and we had no problems.
Then they moved. I didn't think about it because they moved in early winter and I didn't walk over there until early last spring. I had forgotten that there were new people. Nothing had been done to the yard. I had cut across and gone over and cleaned my moms house. I was coming back, tired and out of breath after all that work and was halfway through her yard and she came out and started screaming and throwing stuff at me and saying she was calling the police I was trespassing etc. Of course I froze in place when she started that and when she would stop to get a breath I tried to introduce myself, apologize, tell her that I had forgot they just moved in and the old owners used to let us cut across, but everytime I would start she would scream louder and say meaner things. Yes, I was in the wrong for not asking, but I had forgot, and she had no call to be that ugly. We are the only people who ever cut across there. So finally, I got fed up and shot her a bird and cussed at her and went home. Now I walk around, a block and a half one way, a block and a half the other way. She has a little biting mop dog that chases me. I hope that SOB bites me one day when I'm walking down the back road, because I'll call the law on her so fast her head will spin, that mean old witch! All she had to do was say "Please don't cut across my yard." and I would have apologized and still walked over to where she was and introduced myself and chatted a minute, because that's what nice people, normal people, do. She's not normal. Or nice.
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I'm giving it another shot. We will see.
My forum is still there and everyone is welcome to come join as well. There is a private women only subforum there if anyone is interested. Also, there is no CAPTCHA.
The link to the forum is
http://www.rightplanet.proboards.com