Regulating a LiMnO2 battery at 3V
The way I see it, you have 2 options here. You can either find a new LED with a lower Vf or you can get a boost regulator to bring your voltage up to a 5V rail. There are quite a few low power ones out there, and if you want to save more power you can toggle an enable pin on and off on most devices.

This constant 5V (or whatever you decide you want it as) will ensure your LED does not dim. Even when your battery drops, most of these low power devices will keep a constant output with very low input voltages, which means it should stay stable all the way through the lifetime of your battery. And if your design is space critical, there are some small package ICs out there with fast switching which should enable you to keep your external components small too.

TI or Maxim do some decent low power chips that should suit your needs. 

Just make sure you look at the efficiency graphs on the datasheets!

EDIT As pointed out by Dampmaskin, there is of course a third option, which is to use a different battery. A Li-Ion battery as he mentioned has a nominal cell voltage of 3.6V which should suit your needs too.