Building a good Steward platform is actually a very expensive affair, a good one would require 12 spherical roller bearings:
http://www.hephaist.co.jp/e/pro/ball.html
Or you could try and build a few of your own, but I would strongly discourage that.
For a Stewart platform you are looking to move in six degrees of freedom accurately to any location, so you would need accurate position control for the actuators.
Voice-coils - again, like pneumatic pistons are a force controlled devices. You only have control over the current, which in turn controls the force, you'd have to put in two sets of sensors for position control - a secondary pick-up for velocity and a linear position sensing device (LVDT / optical etc) to do it properly with feed back control through the arduino, you'd also need to have reasonable expertise in control theory to do it right. Also, you must remember that the faster the coil is moving the more back-emf it is generating - i.e. the coil is ITSELF generating power because it is moving. The back emf is dependent on the velocity of motion, and you'd have to account for it in your circuit as well as in you control algorithm. Although only a single optical linear pick-up should be OK, but you would not have sufficient "gain and phase margins" - control jargon for "you wont be able to respond quickly enough to changes". And NO, a "simple PID controller" would NOT work.
With regards to your second question of building a voice-coil actuator yourself: Hmmm ... easier said than done, some commenters have suggested a PVC pipe a bunch of Neodymium magnets - if it really was that simple people would not be selling them for 100s of $, and there wouldn't be people willing to pay those 100s of $ for them!
First, Newton's third law: For every action there is an equal and opposite reaction - i.e. for every action force, there must be an equal and opposite reaction force. For the force that the voice-coil would be exerting, there would be an equal and opposite reaction force. This force is actually acting through the coils, this is where Lorentz force comes in to place. The reaction force experienced by the coils is actual radially OUTWARD.
The trick to making voice coils is getting the winding of the said coils right by balancing its current carrying capacity with the winding tension, the heat carrying capacity of the substrate (no PVC would NOT work) on which the coil is wound with the mechanical and electrical properties of the coil and its geometry. So, yah! Muchos experience in doing helps.
There is also the issue of mechanical integration - there is no real bearing element between the outer sleeve and the inner moving sleeve - if you don't put in some kind of sleeve the coils keep rubbing, get worn out and short (been there don it). There is also the extraordinary risk that some magnetic element might "fall in" between the tiny gap that exists. Now your voice coil is USELESS. You'd NEVER be able to use it again since you cannot get the bloody screw / iron filing / machining shaving out (been there don it!). Now your 100s of $s voice coil is a paper weight and quite an ugly one at that!
Bottom-line: you are better off using a stepper motor to move a screw rod, you only need to know "home" and then keep track of the number of pulses you send out for position - no feedback control - only feed forward. Of course if the loads are too much for the motor to move, then you would end up with "missed pulses", i.e. you'd send out a pulse but the motor would not have moved. You'd keep having to move back to "home"/zero position to regain your reference, OR put in an optical linear pick-up and put in a very basic feedback control, even a PID would work.
Mechanical integration is not easy with stepper motors either - what are you going to use to prevent the body of the motor from itself rotating if you are using spherical joints? You'd need to "anchor" the body of the motor with linear bearings to the linearly moving element.
BTW: If it is not to much of a secret, what is your application for the Stewart platform?