Carolina, here are some techniques that have opened up opportunities for me to turkey hunt:

Join the NWTF and participate - e.g. teach turkey calling seminars at the JAKES events.

Enter local calling contests and become a familiar face.

Volunteer to help out your DNR with wildlife projects (trap and transplant, controlled burns, etc.)

Become a field rep for a turkey call maker and help him out.

Make your own turkey calls.

Give seminars on turkey hunting.

Knock on doors in February. Just seek permission for yourself for a day or two. Offer to help the farmer out (I have picked up trash, baled hay, cut tobacco, chain sawed trees in pastures, bushhogged, etc. - you get the idea, it doesn't have to be $$)

Offer to take the landowner out and teach him to hunt (got my best turkey hunting partner and some of my most enjoyable hunts this way). Share the game with the landowner. Offer to take the landowner's kids out and teach them to turkey hunt. Let someone else pull the trigger. Take pictures and share them. NOTE: Wait several years before even ever considering asking the farmer if you can bring someone else on the property.

Treat the land and landowner with respect. Visit them during the non-hunting season, and remember them during Christmas, even if it's just dropping off a card. (I had a farmer who decided to give up farming on leased land because of his arthritis - he told the new leasee about me, I still hunt there; another farmer decided to lease his land to a hunt club - he stipulated in the lease that I could still turkey and bow hunt - you do not develop these lifelong friendships with $).

Become a state hunter safety instructor, teach the ethics and hunter responsibility portions.

Offer to guide folks you meet who have land: make it fun and safe, not a contest...

Buy your own land and manage it to improve the brood rearing habitat. (The hens in the spring will know good habitat when they see it! The gobs know where the hens hang out...)

Hike in the national forests, scouting where the crowds can't bring their ATVs.
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It can be done, it takes time and effort, but I have more land than I have time to hunt...
-fsh