What is the simplest way to generate PWM without a Microcontroller?
I recommend the (GASP!) 555 Timer in "astable" mode. You'll find everything you need in the link, but I copied them here just for you!

Astable mode gives you a variable PWM frequency, and allows for an adjustable duty cycle as well (high-time and low-time equations in the link).

The circuit:



Note: I'd add an electrolytic cap across Vcc (positive lead) and GND (negative lead) to reduce the effect of dips in power supply voltage.

The PWM frequency:



Some defense for my answer compared to others in this post. Most other answers require an intermediate waveform to generate a variable PWM signal, such as the common triangle wave/comparator method. I don't see much point in constructing triangle wave generator (a significant circuit in-and-of itself) just as an intermediate step to solve your problem.

The 555 is a great analog chip and does just what you need. I wish people didn't hate on them as much.