I'm building my own fixie (almost a cliche, sorry) and I debating whether or not to buy chain tensioners. I will be riding my bike with a fixed gear most of the time. One of the reasons I ask is because I see a lot of fixies without them. So I was wondering whether or not I really needed them. Personal preference is welcome.
If you have an elliptical bottom bracket you won't need a tensioner. Ditto for dropouts that allow horizontal movement.
If you are very lucky even a vertical dropout frame won't need a tensioner but as chains stretch over time this probably won't hold true for long.
When the chain is too short you will have parts of the crank rotation that are very stiff. When the chain is too long it risks coming off and jamming the wheel causing a dangerous skid. In this case a chain tensioner (which could be an old junk derailleur) is a good idea.