For those of you unfamiliar with Churning, it's the practice of signing up for a rewards credit card, spending enough with your everyday purchases to get the (usually significant) reward and then cancelling it. Many of these cards are cards with annual fees (which is commonly waived and/or the one-time reward will pay for). For a nominal amount of work, you can churn cards for significant bonuses.
Ordinarily I wouldn't come close to spending enough money to qualify for many of these rewards, but I recently made the Giving What You Can pledge. I now have a steady stream of predictable expenses, and conveniently, GiveWell allows donations via most any credit card. I've started using new rewards cards to pay for these expenses each time, resulting in free flights (this is how I'm paying to fly to NYC this summer), Amazon gift cards, or sometimes just straight cash.
Since the first of the year (total expenses $4000, including some personal expenses) I've churned $700 worth of bonuses (from a Delta Amazon Express Gold and a Capital One Venture Card). This money can be redonated, saved, spent, or whatever.
Disclaimers:
1. I hope it goes without saying that you should pay off your balance in full each month, just like you should with any other card.
2. This has some negative impact on your credit, in the short term.
3. It should be noted that credit card companies make at least some money (I think 3%) off of your transactions, so if you're trying to hit a target of X% to charity, you would need to donate X/0.97, or 10.31% for 10% to account for that 3%. The reward should more than cover it.
4. Read more about this, including the pros and cons, from multiple sources before you try it. It's not something that should be done lightly, but does synergize very nicely with charity donations.
If you don't want to bother signing up for a bunch of cards, the US Bank Cash+ card gives 5% cash back for charitable donations, up to I think $2000 per quarter. This is a worse percentage but lower-effort and does not ding your credit (as long as you don't miss payments, obvs).
Also, as I understand, it's actually better not to cancel the cards you sign up for (unless they have an annual fee), because "average age of credit line" is a factor in the FICO score. Snip them up, set up auto-pay and fraud alerts and forget about them, but don't cancel them.
It does not seem like the expected value of the probability of something slipping through the cracks would pay for the marginal increase in the credit score.