Sandpaper, maybe 150 then 220 then 320 then 400 then 1000 then 2000.. That's silly, you don't need to go that high, just stop when it looks good enough for you (probalby 220, skip the rest).
You can also "flame polish" the edges by heating them with a torch, enough so just the edge melts a wee bit then cools to a nice clear rounded finish. This takes a bit of practice, since you can easily start burning the acrylic, or melt a bit on the face of the piece.. Once you get the feel of it, it works well. I've used a propane torch with a really sharp flame, and basically quickly skimmed it down the edge.
Ultimately, though, I'd say a piece of relatively fine sandpaper and about a half hours worth of work would be the best.