I was using standard blue painters tape (in rolls) with stick glue on top.  I notice this tape didn't stick nearly as well to the bed as the square/precut pieces that came with the printer.  I actually bought a sheet of BuiltTak after posting the original message.  It works better than anything I've tried, but still get *slight* corner lifting, and the print becomes so hard to get off that I inevitably scratch up the build surface.  I guess maybe a hair dryer or something would help get it off?  I resorted to just building ears into my model to hold the corners down. 

My printer is only a month old, so I don't think it should need greased already (or maybe it just wasn't adequately greased from the factory)?  My first prints were with the clear/nature PLA that came with the Jr, which make it a bit harder to see the banding, but I think the banding was present on those prints as well.  I will have to look again.  Attached is a pic of another print that shows the banding.  Is it the Z-Axis causing the layers to "squish" which creates the banding?  Attached is a pic of another object (printed upside down from the way it is pictured).

I used the blue standard painter tape (3M) and indeed it did not stick very firmly on the bed.
I try ordinary painter tape (the yellow one) and they stick better on the bed but you have to sand a little bit the surface as it is glossy and the PLA is not attaching. After the sanding, the pla is sticking so well that it is sometimes a real pain to detach the object. You should have a try with ordinary painter tape with little sanding on the surface.
As I only print small object, I prefer the Blue tape after all !

Some banding is normal. Buuut... you might need to put some oil on your Z leadscrew. I use Tri-flow oil with PTFE, it works rather well.

You might also need to regrease your linear rails... I use superlube grease with PTFE for the X and Z rods, and since the y axis is kinda hard to get to, I just use the Tri-flow with the extension tube to apply some grease on those.

As far as your corners lifting... It is indeed a adheasion problem. Are you using the stock painter tape squares? Because if you are... or are using painters tape in general... you can also put a additional layer of stick glue "like the stuff that goes on purple and dries clear" on top of that and the adhesion will work almost to well. XD

Another option... you can get something called BuildTak, just make sure you get the z-offset right before printing.

I was using standard blue painters tape (in rolls) with stick glue on top.  I notice this tape didn't stick nearly as well to the bed as the square/precut pieces that came with the printer.  I actually bought a sheet of BuiltTak after posting the original message.  It works better than anything I've tried, but still get *slight* corner lifting, and the print becomes so hard to get off that I inevitably scratch up the build surface.  I guess maybe a hair dryer or something would help get it off?  I resorted to just building ears into my model to hold the corners down. 

My printer is only a month old, so I don't think it should need greased already (or maybe it just wasn't adequately greased from the factory)?  My first prints were with the clear/nature PLA that came with the Jr, which make it a bit harder to see the banding, but I think the banding was present on those prints as well.  I will have to look again.  Attached is a pic of another print that shows the banding.  Is it the Z-Axis causing the layers to "squish" which creates the banding?  Attached is a pic of another object (printed upside down from the way it is pictured).

You might also need to regrease your linear rails... I use superlube grease with PTFE for the X and Z rods, and since the y axis is kinda hard to get to, I just use the Tri-flow with the extension tube to apply some grease on those.

As far as your corners lifting... It is indeed a adheasion problem. Are you using the stock painter tape squares? Because if you are... or are using painters tape in general... you can also put a additional layer of stick glue "like the stuff that goes on purple and dries clear" on top of that and the adhesion will work almost to well. XD

Another option... you can get something called BuildTak, just make sure you get the z-offset right before printing.

Also, for anyone who has viewed the pic in the OP, is that amount of banding normal for the Da Vinci Jr?

As you mentioned about the tape being pulled from the bed, the underlying cause isn't getting things to initially "stick".  From one post I read, the real issue is the expansion/shrinking of the PLA during the printing process.  As the PLA cools, it wants to contract/shrink.  This ends up pulling on the rest of the print.  Corners are particularly problematic under these conditions.  But, I've also had problems when printing circles/discs, too.

The post mentioned "fixes" like using a heated bed (not present on the Jr), modifying the model/G-code so (and I'm not entirely sure how this would work) layers at the edge of the print could be skipped and trying to print solid parts with the least amount of material.  Since I'm not sure about how to do the layer skipping suggestion and I'd like the most freedom to print "thick" things, I was thinking my next search would be for a stand-alone bed heater either specifically designed for 3D printing or something that could be re-purposed for the task.

I don't know if that's much help.  I'm interested to hear if anyone else has found some good solutions to this problem, too.

Has anyone found a method to prevent this with the Jr without printing brims or rafts?  The part i'm trying to print now is too big to print a brim using Xyzware, so I will likely have to build ears or something into the model to try to hold the corners down.  Should I just accept this as normal and use brims/rafts/custom modeling as the solution?