That is what I did, and it works out great for me. Just needs a little lube on the chain after a ride in the rain (like yesterday). My bike basically looks like the silver Bryant above, but has the specs of the Kona below. The only Fixed Gear drawback is the need to carry two wrenches (to prevent axle spin) to change a tire.

Having an inexpensive 8.5kg bike to commute on is nice too (30-35km/hr crusing speed), along with the real silence of a Fixed Gear drive train.

For lights, I have little blinky lights that basically last all season with a dime sized battery, and then USB recharbable LEDs for the times when I need to be extra bright.