What's the most cost-effective service for calling landlines while travelling?

I personally use Skype for this purpose. Their rates are pennies a minute, and they have apps for many different platforms. Google Voice is even cheaper in many cases. To give any more recommendations than that, we would need to know exactly what countries you plan to call, as rates vary significantly. If you plan to use the service frequently, some offer unlimited calling for a monthly fee, which could work out in your favor.

However, for credit card problems, many credit card issuers will permit you to call them international collect, which prevents this problem entirely. Collect calls aren't particularly common these days, but if you google "collect call [name of country you're in]" or contact an operator, you should be able to place one. Credit cards issued by a US bank very often have a number on the back marked "call collect" for this purpose. Visa provides toll free numbers and an international assistance number, which you can call collect. MasterCard has a similar service. Bring a copy of these international assistance numbers with you, and you can get in touch with the credit card companies anywhere in the world.

(Pro-tip: as Visa notes on their page, some hotels may charge their own fees for collect calls, so be careful before trying to do this from a hotel phone.)

Skype with a bit of credit is an excellent approach- you can call any number in most countries and I believe that 800 numbers in the US generally work (that becomes important when you only have an 800 number- it won't save you money). A half hour call from China to Canada, for example, is something like a dollar. Some countries rip you off with extra fees (eg for mobile phones) but these are just the countries that will make it illegal to use VOIP, and may block it. Some countries block Skype too, but you can often get around that with a VPN.

Aside from emergencies, some folks are just more comfortable using a real telephone- they don't have to know what is going on at your end, so the credit can serve both purposes.

If you install the app on your phone and computer you can make calls wherever there is reasonable (often free) Internet. I was in the arrivals lounge of an Asian airport last fall and a man was at the pay phone going ballistic because it was costing him $50 or $100 US in long distance fees to work on clearing up some kind of a travel snafu. I have cleared up a number of such problems with little extra stress and almost no unnecessary cost using a combination of IM, Skype out, email and (when justified) a local SIM card.

I do have VOIP phones and could, in theory, use those directly, but my experience in getting them to work reasonably well requires using a SIP server that is near to your current location. So I would have to have a list of sites and change the settings on my VOIP software (I have Voiper). The rates are cheaper(maybe 2/3), but unless I was spending days on the phone it would make little difference and the buggering about has been non-trivial.

If you need to receive calls then it can get more complex/expensive, but I have never felt that need. E-mail or IM is typically enough, and even without a local SIM a text or two won't typically break the bank.

The above are suggestions that have worked for me in many countries, however they probably won't work in some places- I would guess Iran, North Korea etc., for political reasons, and places with dodgy Internet, for technical reasons.