Building a Circuit that takes constant mA
The most you should discharge a Li-ion cell to is about 2.8 to 3.0V to prevent damage to the cell.
Here is a very simple circuit that maintains a fairly constant discharge rate from 2.8 to 4.2V (and beyond).



simulate this circuit &ndash; Schematic created using CircuitLab
R2 biases D1 (actually an IC shunt reference that regulates at 1.24V) and provides a small amount of base current for Q1. Q1 provides the remainder of the base current Q2 requires.
R1 is chosen so that 1.24V/R1 ~= 76mA. A bit of current that is not metered goes through R2 so the current is not precisely constant.
Here is a quick LTspice simulation:

The circuit will typically work well below 2.8V-3.0V so you need to provide some means of cutting off the current before the cell is damaged.

Just for comparison, as I already had it set up, here is @Michal's circuit simulated:

It's possible that he got different results depending on the models used.

In real life, it would be much nicer to make a circuit that operates from an independent regulated power supply such as 5.0V which would eliminate both the low voltage limit and most of the slope in the current-voltage line. It would also be easy to incorporate a low voltage cutoff.