I know exactly what you're talking about. From looking at the pics, it looks like he made the staggered look by taking 1/4" and rounding off the edge, then bonding that to another piece that had the same done to it, then again. Sound right?
If that's the case, then you have a routered roundoff and that is the part that you are talking about. The first thing I would do it try to take off the router marks with a fine grit wet/dry sandpaper and this is where taping off the finished area around it will probably help. Start with 1200 or 1500 and a soft block to hold, keep the area you are sanding wet by spritzing with a spray bottle and also spray the sandpaper also. It helps to have a couple small buckets of water to dip the sandpaper into, one to get the dust/particles off mostly, and other to get the rest off (these can be small containers). This keeps the sanded particles from making more scratches.
You can probably go for 30 seconds to a minute between 'dips' depending on how fast you're working. you shouldn't need to work an area very hard to get the router marks mostly removed.
At this point, you can probably flame-polish it with MAPP gas and a flare nozzle, making passes as moderate, even speed (this takes some getting used to), but if you haven't done it before, practice on some scrap. If you mess up, you can sand again, etc, but eventually there might be no going back. Not to scare you, just sayin.
The other option is to buff it. Rinse the acrylic and use a buffing wheel with an arbor bit on a drill. You can get these at most hardware stores. Look for a 3-stitch wheel (it is a bunch of cotton circles stitched together with only 3 lines, others have many more stitch lines, you don't want those. Get some plastic polish, something like this
http://www.meguiarsdirect.com/product_detail.do?q=4729
http://www.meguiarsdirect.com/product_detail.do?q=4814
The first will take out the 1200 sanding marks, the second will polish to a mirror shine. You will want to use separate wheels for each compound.
This is what one company in CA uses (featured on LA fishguys recently) to polish tanks. You can get cheaper stuff, most 'final' car polish will work since this isn't in the tank.
Anyways, Spread a very thin layer of it on the acrylic (leave the tape on, or likely replace the tape) and let it dry a little, then run over it with the wheel at relatively high speed (2000-3000 RPM for a 4 or 6 inch wheel even). You won't need to press down on it, just get the wheel close enough to touch the acrylic. Oh yeah you will want to secure the piece to something sturdy before buffing. You will probably want to re-apply the polish several times as the wheel will soak it up. Anymore I drizzle it right on the wheel edge and even it out with a paper towel or cloth, but when you spin it up it will spray that stuff everywhere so be careful (eye protection).
After you get the hang of that, you'll probably be done with the ultra-cut step. It won't be finished yet, but it'll look a lot better, and you might be happy stopping there. It may still have a dull look to it yet. Go with the 205 (ultra finish) and you will likely get very close to a mirror finish. Then, if it's still not to your liking, a couple quick passes with the MAPP gas should make it shine up really nice.
HTH
Bud