I've come up with an approach that I modified from several different people who've been pretty successful at getting into private yards. It goes like this: "Hi, I'm (whatever your name is) and I live over by (a local landmark if it's the same town or just the name of the town if I'm in a different town). One of my hobbies is metal detecting and I mainly look for old coins like these (I carry a pill bottle with some various silver coins I've found....enough to get their interest). I've had pretty good luck around homes like yours and I was wondering if you'd let me hunt your yard?"
That brief opening is usually enough for me to tell if I'm going to have a chance or not. Some people say yes right away; some say no. Some are on the fence though and you'll need to be quick on your feet to close the deal with them. If they seem like the type that might go for a chance at getting something out of it, I'll add: "You know sometimes I do find some gold rings and stuff, in fact, I found a ring worth $650 just a few weeks ago (don't make it up though, be honest). If I'm lucky enough to find any gold in your yard, we'll split the value 50/50. Either you can keep the item and buy my half or I'll keep it and pay you half the value. I am a coin collector though, so I want to keep any old coins I find." That extra part will usually get you into the yards where the owner is on the fence.
When I go to the door, I don't have any MDing tools. I have on clean clothes and am well groomed. You don't want to look like a bum or have things that might intimidate them hanging off your belt. Be courteous, even if they say no. I've had a couple of occasions where the person said no at first and after I thanked them anyway and started to walk away, they changed their mind and said go ahead. Take good care of the yard and don't do any damage. If you find some interesting stuff, show it to them, but keep it in your hand! If it's a really good yard and you find a lot, offer them a silver coin or something that you can tell they have an interest in. Word will get around to other neighbors, either good or bad, about how you conduct yourself.
Good luck. I was a shy boy myself, but after 30 years in the military I could talk at a Congressional inquiry now! LOL.