Use the 2nd circuit in post 32.
Q1 and R9 are the only 2 new components. The rest is just so I could simulate your circuit.

Gotcha. That was basically my plan for today. I'll let you know how it goes, and thanks in advance!

 

For posterity sake, this is what I ended up building:
http://goo.gl/pdD2NP

I figured out that the "sweet spot" for the resistor/potentiometer feeding the inverting input of the third op-amp was about 10k - anything more, and the circuit didn't have enough current to work. Since I wanted a fixed output (instead of using a potentiometer), I went with a 4.7k resistor. However, using such a small value meant that the ramp sped up rather quickly. In order to slow the ramp down, I used a larger 68uF capacitor, and that works quite well.

I certainly learned a lot from this little project, so I thank everyone for their patience and willingness to help me learn!