The setup you describe is often called a 1-by- configuration, as in "one-by-seven" or "one-by-eight". I just made one for myself for commuting and light touring (cyclocross fun in my future??). The two issues you're likely to encounter are chainline and chain dropping. The front derailleur does act as a chain retention device when you are dealing with rough terrain, weather, and front shifting under load. If you find our chain frequently dropping, you may need a chainring guard, which is like an outer ring with no teeth, or a chain deflector, which keeps the chain from falling inside.
The other issue is that many single speed cranksets are meant to keep the chain very close to the bike, in line with your freewheel/cog. If you start using a 6-speed freewheel, by the time your chain moves all the way out to your smallest cog, it will be very crooked (cross-chained). This not only increases the chance of your chain being dropped, but will also cause it to wear out faster. I dealt with this issue by using a longer bottom bracket and the middle position of a triple crankset, which sets the chain slightly further out than the inner position on a double crankset (in both cases the outer position was used by a chainring guard. I wish you the best of luck.
1-by bikes are not very common for several reasons. First, more speeds is often a selling point by shops and manufacturers; any X-Mart bike is going to have "21/24/27 Speeds" as one of its top selling points. Second, the parts market, both new and used, is saturated with multi-speed cranksets and cheap front derailleurs; oftentimes a you can get a triple
for less than a single
. Front derailleurs and their shifters are less complex than rear derailleurs, and thus add negligibly to the cost of a bike (especially since retailers often spec cheap/lesser front derailleurs than rear ones).
That being said, there are several types of bikes that oftentimes are run as 1-by's--those being cruisers
, light duty city bikes
, and cyclocross racing bikes
. I personally just finished a 1x8 commuter/cyclocross bike myself. The front derailleur does help hold on the chain in bumpy conditions, and thus cyclocross bikes as a rule have chainring guard/chain deflectors installed to prevent the chain from falling off. On city/cruiser bikes, the lack of front derailleur makes pants guards/chain guards easy and practical to install, and these also generally do double duty as chain retention devices.
My personal decision to use a 1x8 setup came from over a year of analyzing my own riding habits. Since I was in the market for a new commuter anyways, I decided to build one up with just the gearing that I used. So far I've put over two hundred miles on it, including one weekend extra-light tour (25lbs additional gear, 90 miles round trip) in the Texas hill country. Thoroughly enjoyable so far, and the fact that I "upgraded" from a road bike to a cyclocross setup certainly helps (rediscovering the joys of jumping curbs). I say do it, ride how you want, and you can always easily switch back to a multi-speed crankset later.