Arduino-like boards with other microcontrollers
You have to define what you mean by "Arduino-like" beyond just not using an AVR.

There's the Arduino Galileo using an Intel chip and supports the Arduino development tools, as well as the Arduino Due which is an Arm-based chip.

There's the Netduino which uses ARM chips (STMicro's) and is pin compatible with Arduino shields, but I don't know what the development framework is (other than it being in .NET), though this doesn't necessarily say much because not even all Arduino boards have the same pin compatibility (Nano boards are much smaller, LilyPad Arduinos which are round and semi-flexible, etc.)

Then there are related boards, but don't support the Arduino environment or any pin compatibility (TI Launchpad springs immediately to mind, but pretty much every manufacturer has some sort of demo board or development board for their MCU's).

There's a huge list of Arduino-like boards here (split into levels of compatibility, includes the official Arduino boards).