To tell you the truth, we normally just use a pair of tin snips. I’m not sure if this is the best way, however. 
~Jordan
We use a sharp utility knife to cut, similar to what is shown here: http://www.youtube.com/watch?v=MeAbhAK9luw
You could use a scroll saw. This would work for both straight and curved lines. My dad didn’t have a scroll saw, so we used a dremel for the curved lines. It was not as smooth as using a scroll saw, but it worked.
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You can also go “old school” and use a Coping Saw. Back in the dawn of time they were used to make intricate cuts. Small, portable, quiet and inexpensive. See your local hardware store.
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I just used a box cutter now, like recommended above, and it worked well for the straight line cut I needed to make. Had to go over the line repeatedly, though, to make the cut deep enough.
As LegoMindstormsManiac said, tin snips and/or scissors AND a lot of time :P.
We usually don’t go beyond basic geometry – squares, rectangles and triangles usually. I’m curious as to what 1107b uses to cut theirs… they are the lexan gods.
my team uses a jig saw, dremel, or coping saw. To keep it straight, we just use a corner of a table or a ruler(metal) as a guide with clamps. After that we just file it down to get it nice and pretty. If you want sharp curves with the jig saw, just add relief marks.
We use a laser…
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Where do you buy TADADADA’s? I haven’t seen those in the local hardware stores here in NZ.
We just use tin snips too. Sometimes the edges are a bit rough, so (if I can be bothered) I sand the edges and corners.
~George
This is what we used until we broke the handle. LOL. 
Is that with or without a shark?
I like your style.
We use that or waterjet and if its thick enough (college) we use mills and lathes.
I like the look of waterjet Lexan though, its pretty cool.
But you can use hacksaws, utility knives, band saws (horizontal and vertical)
Machining is fun, because there are SO many different ways to do one thing.
- Andrew
Last year we just used an oscillating tool. We just had to sand the edges down after each cut to make it look really nice! It’s good for almost any cut.
I believe it could also be with a cat…
My team uses either a band saw or a dremel, depending on what we have access to and how soon we need the component.
+1 for the tin snips for straight-ish lines.
Additionally, if we needed a fast cut, we’d hack-saw.
Finally, reading through the responses, I feel like you wanted something “intricate”, the cheapest route would be to trace it out and then run over it with a box cutter as one poster suggested.
- Sunny G.
If no scroll saw or band saw is available, next choice is a Jig Saw before coping saws or tin snips. Clamp it down and use the right blade. Some models even have freakin lasers on them. Not on their shark heads per se, just lasers…
This is the pretty cheap and handy Black and Decker model line up…
http://www.blackanddecker.com/allproducts/power-tools-jigsaws.aspx
We always use a hacksaw to cut polycarbonate. It’s quick, and all you need is a quick Sharpie mark before to mark the lines and a bit of filing afterward to give a clean finish.
All these fancy tools, whats wrong with scissors? That’s what we use 90% of the time, and then a ban saw when we are lazy =P
Jack, was it you that broke our pair of scissors when trying to cut polycarb?
We find tin snips work fine, we once tried using scissors but we now lack any of those, but perhaps a stanlely/craft knife would also work fine. That way you could get a fairly straight clean edge easily.
Yes… I was hoping you wouldn’t bring that up on defends they were little kids scissors… we haven’t broken any others (yet).
Tin snips, maybe we should get some then