I always ask in person. Like others said, be personable, shake hands and be polite. Dress presentable, you dont have to wear a suit or anything, but dont show up in a ghillie suit or hip waders either.
Where I live I usually have no problem getting land to trap or predator hunt. Deer hunting is the opposite; I get about a 10-15% permission rate. Turkey runs about 30-40% in the fall and 50-60% in the spring. My biggest problem is I move too much. Develope a report with landowners and respect their wishes and land and you will find many will let you hunt, trap or whatever later on. I think the biggest reason my deer hunting permission rates run low is not because they hunt the land themselves, but because other people ruined it for me. Many will tell me right out that someone shot near their house, shot their dog a few years ago, rutted up their fields, things like that. Many do hunt deer on their own land or have family/friends that hunt it, but not like many people think.
I made up a brochure type thing for myself and I have found that landowners are impressed with it. It has all my information on it, my name, phone number, address, vehicle (including license #), etc. I also put the fact that I have 18 years of hunting experience and 10 years trapping experience. I put that I have taken a trapping safety course and hunter's safety course (both are required in Wisconsin, but not everyone knows that unless they hunt or trap themselves). I am state certified in firearms as part of being certifiable in law enforcement so that's in there. I put the clubs I belong to (for me Wisconsin Trapper's Association, National Trapper's Association, Fur Takers of America, Ducks Unlimited and Whitetails Unlimited). Put anything that shows you are a person that knows what you're doing. They like to know that you know what you're doing and they dont have to worry about you shooting their pets or livestock or shooting at their house.
I have in mine the Wisconsin statute that protects landowners in case I am hurt or anything I do. Wisconsin's statute is 895.52 and I have that spelled out for them. Not all landowners know that and fear if I am hurt they would be liable for me since I'm on their land. I am sure other states have similar laws so check it out and put it in there.
Then I have reasons to allow trapping and hunting on their land. I spell out the whole carrying capacity theory about disease and starvation etc. I put in there a little on how today's traps work. Keep in mind that unless people hunt or trap themselves they may not know how it works. Some people think traps are barbaric torture devices with teeth etc. I have in there that traps with teeth are illegal and were retired decaded ago. I have in there that traps are checked every 24 hours (Wisconsin law and my preference) they may think (as many do) that an animal held in a trap is sitting there writhing in pain for days and/or starving. Educate them a little. It's nice to have the brochure for people who aren't sure or don't want to commit to a yes right there on the spot. I leave the brochure, they can read through it when it's convenient for them and they then have my contact information to get back to me with an answer later.
Once you get permission, respect it! They pay taxes on that land and are generous enough to allow you on it. I personally hate hunting or trapping public land (too much hunting pressure, trap theft etc.) so I greatly appreciate getting on private land. Don't drive where you aren't supposed to or create ruts in their fields. Obviously don't leaver trash behind. Dont put your carcasses on their land for their dog to drag back to the house. I find a few that will allow me on the land only at certain times (especially during deer season). Respect that. Dont try sneaking in and playing dumb. Basically use your head.
Once you hunt or trap on their land for a season or two and they see you are respecting their land you will have a spot for years to come and possibly may find they'll let you hunt deer or turkey later on too. I recommend you make something similar for landowners, it works out great.