Tom,
When someone mentions oily wood, the first one that comes to mind is cocobolo. The general consensus among experienced segmented turners is DON'T USE IT. I guess I didn't know better and hadn't talked to other segmenters when I used it in some of my segmented vessels. Fortunately, the pieces that contained cocobolo segments have held together so far, at least I have not heard any complaints from those who own them. I am showing a picture here of a lid for a vessel that has twelve segments in the rim surrounding a mesquite burl. It was made seven years ago. You can hardly see the seams and the surface is perfectly smooth without "glue lines". Now this lid is only about 3" in diameter which may have helped it hold together. Over the years, I have used Titebond, Titebond II and Titebond III for gluing my segments. Unfortunately, I have not kept a record of which glue I used on which piece. However, I have never had a failed joint, although you can feel the glue lines on some pieces. I only recently noticed that Franklin, maker of Titebond, claims in ads that Titebond would work on oily wood, but I can't substantiate that. I don't believe in wiping oily wood with acetone and I am skeptical about the notion that the freshly cut oily wood will glue but if you let it sit it will fail.
Cocobolo is such a beautiful wood that it is tempting to use it in segmented piece. However, it darkens relatively fast. The lid in the picture has been kept in a cardboard box (from light) most of the past seven years but it still darkened to the point that you can't tell it is cocobolo. There are many other species that will give you this dark accent but without the concern of failed joint due to the oil content. My advice is to stay away from it.
Sorry I didn't chime in sooner. I was trying to take a picture for the discussion.