I got the Octahedron II a while ago and I finally worked out a solution for it and solved it. ETA: The Rhombic Dodecahedron is essentially the same puzzle, so it belongs in this discussion.

Comparison of the Octahedron II (O2) and the Rhombic Dodecahedron (RD)
The stickering of the RD is much easier for me to follow, as the pieces (without giving anything away) are colour-grouped more logically. One piece on the RD is a tetrahedron (3-sided pyramid), where corresponding O2 piece is a flat triangle. This fits the colour scheme as the RD has three colours on this piece, while the O2 has only one colour.

The two puzzles will be solved identically, with reference to the pieces. Unfortunately, the grouping of colours will be different. I have determined that the stickers sets for the RD will fit the O2, but as this puzzle was deliberately designed for it's colour scheme, I will respect this design intent.

My solution to the Octahedron II is based on my observation that the Octahedron II is a
[SPOILER]Shape mod of a 4x4 supercube. The "points" are actually centres, the rectangular pieces are edges and the flat triangular pieces are corners. The stickering makes this puzzle even more difficult as the Octahedron II is divided into octants of a single colour, comprised of a corner and the adjacent pieces (edges and centres) on what would be three 4x4 faces. The octants are easy to move around the puzzle and you could treat them like the corners of a 2x2 cube.[/SPOILER]

My solution ideas:
[SPOILER]I initially approached it like a 4x4, though I did struggle with edge pairing. I now understand that edge pairing rotates the centres which, on a regular 4x4, is not apparent and not significant. I developed some useful algorithms to cycle edges and centres. I also found some supercube algorithms to cleanly rotate the centres. I know that I can solve it, but I'm still working out a good sequence (e.g. group centres and then pair edges or pair edges and then insert centres). I should note that I did quite well in assembling the first 3 octants by intuitive methods, but after that there was no free space to assemble them.[/SPOILER]

I'm interested in your ideas and analysis for these pleasant and inexpensive puzzles.

This is a little update on the Octahedron II. I can't overstate how difficult the stickering makes this puzzle! The centres and edges have four and two different colours respectively. This means that centre assembly and edge pairing involves putting together quads and pairs of essentially unrelated colours. The colour relationships start to form after the centres and edges are assembled. I have devised a colour plan, but I'm not sure I will be able to do it from memory.

I have learned a lot about the algorithms that I've been using for the base puzzle. For example, the edge pairing algorithm I use leaves the U layer rotated 90° CW and flips a U-layer edge pair. It's not an issue when you can see it (and have one-colour centres), but in a rainbow of eight different colours, it is easy to lose track of a rotated centre. I think that this sort of puzzle makes me better at solving all puzzles because it makes me think - really think - about each algorithm before I apply it and check that it did exactly what I expected and nothing I didn't expect.

BTW, the Octahedron II is inexpensive (less than $20 shipped). It is listed as "4x4x4 Octahedron II" on HK NowStore, but as "Rhombic Icosahedron(Scopperil)" on other sites. It does have many properties of a rhombic icosahedron, but something is a little off with the shape.

One more thing... this simple little puzzle is so much fun it is probably illegal in many jurisdictions !

Octahedron II Solution Outline:
[SPOILER]
1. Make the centres using 4x4 techniques
• Put 4 different colour centres together (e.g. WRPGy) this is the "white" face
• Put 4 more different colour centres together on the opposite side (e.g. YBOGn) this is the "yellow" face
-> Use cuboid corner swaps on a 2x2 "corner" to get the centre pieces in place
• Pair up remaining centres on the E-layer
2. Pair all 12 edges
• Match up pairs of rectangular edges using 4x4 edge pairing technique:
Top: WR RP PGy GyW
Bottom: YB BO OGn GnY
Sides: BW OGn GyP WR
Note: Use 4x4 edge parity alg to assemble any last edges. Add a final U' to the standard alg.
*** Do not split the centres after this step ***
3. Bottom Cross
4. F2L
5. LL
*** Somewhere in the LL, may have to use a supercube centre rotation fix. Keep checking and fix it if/when it crops up.
• Orient LL edges using 3x3 techniques
• 3-cycle edges using SUNE or ANTI-SUNE
• Fix any swapped adjacent edge parity with the even-order face flipper alg
• Use beginners corner 3-cycle to permute corners (there is no corner orientation as they are 1-colour triangles)

[/SPOILER]
ETA: Updated my solution outline to be more specific about the LL.

ETA2: The 4x4x4 LanLan Sepak Sakraw Magic Cube or as HK NowStore calls it, the Super 4x4x4 Mask Cube II is a more compact puzzle that is functionally equivalent to the Octahedron II and would be solved identically. It might be a little easier to solve, as it more closely resembles the base puzzle. I'm really tempted to get this variant. Looks like I picked the wrong week to take up collecting Shapeways puzzles .

I don't have the Oct II but I do have the RD, so this is my solving outline for the RD.

1. Solve centers (like a 4x4x4, when I talk about centers I'm referring to a completed "center" being made up of the correct 4 pieces which each of which have 2 stickers on them)
a. First I solve the 4 corners (the 3-sided pyramids) on my "top" face (top being whichever face I choose first). I do this not because it's essential but because it helps me know which colours to start solving the centers by. [eg. solve the corners red blue orange, red grey green, grey yellow lilac, lilac blue white.]
b. Solve the top center. [eg. red green lilac blue]. This is done in the same sort of way as a 4x4x4 cube but obviously attention is paid to the colours. Each piece has its specific position.
c. Solve opposite center, again like I would for the 4x4x4.
d. Solve remaining centers as for a 4x4x4.
e. If I get down to 2 or 3 center pieces out of position, I use U r U' l' U r' U' l to pure cycle center pieces UBL->FUL->FUR.

2. Make edge pairs. Done the same as for a 4x4x4, by bringing edges together (disrupting centers), moving them onto a different face and then returning centers.

3. Place paired edges.
a.Same as 4x4x4, and making sure not to disturb the centers.
b. If I need to swap two edges, I do this as I would for a 4x4x4. This process is simple but difficult to explain.

4. Place corners. Using U R U' L' U R' U' L to pure cycle FUL->UBR->UBL.

By far, the most annoying/difficult part of this puzzle is getting the centers. Once they're done, the rest is fine. It took me a fair while to get my head around the solving method to adopt and I sort of avoided this puzzle for a while, but once I started "seeing" it, it made a lot more sense and I ended up enjoying the puzzle.

I did make a video of it all at my blog or youtube.