Best app for texting internationally to non-smartphone mobile?
The app in my mind apparently does work abroad and internationally.
Pinger (free)
Pinger is a cross-platform messenger and a free texting/calling application.  It allows you to grab a US number and use it overseas- without having your family back home paying crazy international fees.
Here is a breakdown of what's absolutely free: texting, messaging other Pinger numbers, obtaining a real US number, inbound calls from anyone, and calls to Pinger numbers.
Outbound calls can be purchased in packs of minutes or by watching short advertisements to gain free minutes.
From Pinger support:

Traveling outside the US
You can still use Pinger! Pinger uses the internet to work. Just
switch your device to airplane mode (don't want to get crazy
international data-roaming fees!). Anytime you have WiFi, be it free
in your hotel or paid in a cafe, you can use Pinger to text or call
your friends at home.
[...]
Will my friends get charged international fees?
Never fear!
Because Pinger gives you a US or Canadian phone number, you can use
Pinger and your friends at home WON'T get international fees when
you're abroad! Only their standard texting and calling rates will
apply. Pretty sweet, isn't it? :)