- heat water to a boil
- add malt extract
- add hops periodically
- cool your wort
- pitch yeast when the temperature is cooler than lukewarm
I'm sure many people have made beer in a similar way and enjoyed it immensely. I doubt my grandfather used a thermometer when brewing for the fourth of July parties back in the day. So, no you don't need a thermometer.
I've made a few batches like this, but, like many people I use steeping grains to improve the flavor and color of my beer. In order to extract the sugar from the cracked specialty grains you would typically use hot water. Not too hot or too long or you might release too many tannins into your beer giving it an off taste. Read more in this detailed article at BeerSmith. So, steeping grains can be aided by using a thermometer to monitor temperature. It will also benefit cooling your wort and preparing to pitch yeast. Depending on the yeast strain and the pitch temp you can affect the flavor profile, kill the yeast or increase the time before active fermentation or lag time.
For years I used a dial thermometer like this one from Northern Brewer (dial thermometer). People enjoyed the beer I brewed with it and it is probably sufficient. On the other hand you might consider purchasing a thermometer that does more than just take temperature readings. I wasn't shopping for a new thermometer, but, last year I happened to see a deal on this digital thermometer at Amazon, Taylor Digital Cooking Thermometer/Timer. In addition to providing temperature measurements it includes a three foot probe, magnets on the back for mounting on a metal surface like the back of the stove, a timer and even a temperature alarm. For around $15 I couldn't resist. I even use it as my main cooking thermometer now.
So while you don't "need" a thermometer most homebrewers would probably say it is essential to making good beer.