The reason I went deep into researching induction (natural, mechanical or chemical) is because with my babies I've had gestational diabetes that tends to spiral out of control at 37 weeks. I've delivered at 41 and 38 weeks, and delivering instead of struggling with ever increasing insulin doses was a better option for us. You should always weigh the risks of encouraging labor vs the risks of continuing the pregnancy. Yes, a baby might be viable at 39 weeks, but in the process of forcing a delivery you may cause a problem that otherwise wouldn't be there (ie, injury to baby, NICU stay, infection, maternal hemorrhage).
Some people argue by doing this you wont be having a natural labor/birth. I get that, I respect that perspective, but if your options are this or a possible hospital induction with Pitocin... these are good tools to have in your toolbox just in case.
So... this is just what I have done, some things I've experienced and learned in my own pregnancies. This is not medical advice, I still suggest you talk to your doctor or midwife.
RRL - I half heartily do red raspberry leaf (RRL) tea throughout the pregnancy. It is supposed to improve uterine tone, so when you are in labor your contractions will be more effective, strong uterus's do a better job at pushing out babies. I like science and studies and statistics, so I don't really know how helpful this really is because I've never found much science behind it, but I put it in my "hey it can't hurt" category. I recently learned you can take capsules and I also heard you can get a tincture which is stronger/more effective.
EPO - At 36 weeks I start taking evening primrose oil (EPO). When I asked my OB about it during my first pregnancy he didn't seem to think it would do any good... but... after reading into it I learned that EPO is high in prostaglandins. I'm sure you've heard to have sex to induce labor, well the reason for that is because semen is also high in prostaglandins (helps sperm accomplish mission impossible). Also, once when I was researching chemical inductions (what you will encounter in a hospital) I learned that the cervix ripening cream/tampon often used, Cervidil, main ingredient that makes it work are prostaglandins - but from pig semen, which I personally find gross. EPO doesn't start labor, it softens/ripens the cervix which is an essential part of birth. Starting at 36 weeks I take 1000mg 3X a day orally and insert one up against my cervix at night. I know some women who use the oil as a lube... double prostaglandins score! (update: I've since read that EPO may increase risks of bleeding, I did not take it with my 3rd of 4th pregnancy.) (update: I've learned that EPO does not have much research supporting it, and may increase bleeding issues. I did not take it with my 3rd or 4th pregnancy)
Sex - Like I said I'm sure you've heard this, everyone loves to suggest it, but sex totally does work. Lots of reasons... because of the prostaglandins, and orgasms cause uterine contractions (gentle, nothing that will start labor but like a little uterus workout) and your body releases oxytocin, which is also released during labor. If you choose a hospital induction you will likely be giving Pitocin, which is a synthetic oxytocin (only instead of gentle contractions it'll make your uterus contract painfully hard and can distress baby). Knowing this, I never let our sex life go during pregnancy, esp the last month. Uhhh am I gonna share this? Yeah... of course I am... we even had sex during labor (my water hadn't broken) and it totally kicked things into high gear, I started transitioning a little over an hour later!
Nipple Stimulation - Nipple stimulation also releases a lot of oxytocin. I know some hospitals even suggest women use breast pumps and have them available.
Herbal Tinctures - During my pregnancy with Ash my midwife suggested a "mother's blend" tincture, which you can get from someone trained in Traditional Chinese Medicine. I wasn't a big believer but I figured, 20 bucks, I'll try it. SO impressed! My Braxton-Hix contractions really would kick up, not in frequency but just in strength. While this was annoying, like labor I knew it was doing something important. The day I finished my bottle, 37w5d, I saw my midwife, asked her to strip my membranes (sliding a finger into the cervix to slightly separate the bag of waters from the uterus, which releases hormones). I went into labor the next morning, it was awesome. This just strengthens your uterus, the tincture I used contained black cohosh, partridge berry, and black haw. Did I mention I walked into the hospital 9cm dilated and ready to push and I had experienced almost no pain? While I partially credit preparing for a natural birth/being informed/relaxation, I think everything I did to strengthen my uterus and soften my cervix made labor a lot easier... if such a thing exists.
Walking - Walk and stay active right to the end. Use gravity, the swaying motion of walking helps baby settle into your pelvis and his or her head will put more pressure on your cervix. I do lots of squatting, like just when I'm playing with the kids and stuff, because that helps baby settle down more also. I also watch tv on one of those yoga balls and bounce. Anything to help to get baby low and ready to go :)
Membrane Sweep - This is the most invasive and some women complain that it causes uncomfortable cramping. During a pelvic exam your care provider inserts a finger into the cervix and separates the bag of waters from the uterine wall just a bit, which causes prostaglandins to be released. If this is your first baby, don't get your hopes to high on the sweep working, it's more effective in women who have already had a baby... but if you feel comfortable with the idea of trying it then it's a great option.
What else have I tried? Everything. But the above are what make the most sense from a scientific and logical stand point in my opinion. I've also drank castor oil (there's a video, I'll post it at the end, castor oil sucks), spicy food, ate a whole pineapple in one sitting (the internet told me to), balsamic oil vinaigrette, date fruits, bananas and acupressure but in acupressure's defense, I think I did it wrong at the time.
Something to remember is that if this is the first time your body has ever birthed, it might not respond, just as it will be less likely to respond to an induction in a hospital. On average, first time moms will go into labor at 41w1d... I know... no one likes hearing that... that's just average and not a death sentence. Ideally you can just enjoy the last weeks of your pregnancy, but I hope you find the above suggestions helpful should you need them.