The worm gear solution described above is possibly the solution with the most positioning accuracy - you can see it at work in homemade CNC machines.
The simplest and fastest solution is to use DC motors to power wheels on a car. The wheels go around, the car moves forward - circular to linear motion conversion.
The next step up from that is a wheel with a pulley, which can be just as fast as a car but not as accurate as a worm when in comes to positioning*. Still, it's the solution I'm opting for in my trombone playing robot to move the slide back and forth, and I'm seriously considering using some small motors with a single pulley in the design of some small actuators.
*Sorry that's not exactly true. A single pulley will never be as accurate or have the mechanical advantage a worm can provide, but many pulleys and a lot of line can. Many pulleys and line can also be much quieter and result in less wear in the long run, whereas all worms inevitably wear and will get noisier as a result. Noise isn't a big deal for most applications, but is in mine. The reason for favouritism of the worm gear is that it provides a lot of mechanical advantage in a very compact form, whereas a lot of pulleys (or, as another example, a lot of spur gears) take a lot of space, manufacturing, maintenance, etc.