I've been growing lemon grass Cymbopogon citratus for about 7 years now. I grow it in the ground and in pots. In fact the ones in the ground lived through 6" of snow last Christmas. This spring I just pulled out the brown parts that were on the outside of the plants. And I have a chef friend that uses lots of my lemon grass at his restaurant.

Anyway, it is common for the outer leafs to turn brown and die off and the new growth comes from the middle. Unless your entire plant is turning brown without any new growth in the middle, don't worry about it. Just make sure that it is in a sunny area, well drained, tends to like slightly acidic soil, and give an occasional dose of fish emulsion and/or sea weed emulsion. As long as you have new growth coming up from the middle, don't worry.