Three inches thick would stretch any router - the longest bearing guided template bits I know of are have only a 2in depth of cut. You can get jigsaw blades to do 4in (Bosch/Metabo/Scintilla), but they are only designed for straight cuts. For detail curve cuts jigsaw blades normally have a maximum depth of cut under 2in. 3in would also stretch most scroll saws, too.
If I were doing this at home I'd use a bandsaw with a 3/16 or 1/8in blade - 2 to 4 tpi hook or skip tooth - to cut the shapes roughly to size. The finish you get off that will require cleaning up, however, so unless these are one-off pieces you may need to clean them up using a router. The way to do it is to attach a plywood or plastic (Tufnol or acrylic/perspex) template to the workpiece (or if these are small pieces, attach the workpiece to a template - so your hands are well away from the cutter) and rout the outside of the shape using an inverted router in a table with a bottom bearing template copy bit (so the "sandwich" is from bottom to top table top/rtemplate/workpiece/workpiece holder). If you use two different lengths of bit (say 1in and 2in depth of cut) you can achieve about 2-1/4 to 2-1/2in depth of cut in about three or four passes. In order to achieve the depth of cut on the last pass it will be necessary to remove the workpiece from the template and use the guide bearing against the already routed edge of the workpiece. (Bear in mind that you may need to make-up some form of jig to hold the workpiece). Finally flip the workpiece over and rout the last 1/2 to 1in of the edge using a bottom bearing router cutter in the inverted router with the bearing running against the already-routed edge. There will be slight witness marks from the multiple passes, but they do sand out.
Whatever you do resist the temptation to take a hand router to the wood, it is much safer when doing these sort of pieces to work with a stationary (inverted) router and make multiple passes to achieve the depth of cut you need. I'll suggest that you need to use at least a 1000watt router to make this sort of cut. Always use a jig to hold the work so that your hands are kept well clear of the cutter at all times. Lastly make sure that you have trimmed as much waste material away from the outline with a bandsaw, chisel or whatever BEFORE making any cuts with a router - this reduces the chance of the workpiece grabbing the cutter and being ruined - or worse still of you having an accident. Grabs are a real danger when deep routing like this.
If you find the task of routing too daunting, a professional furniture-making shop should be able to rout the shapes for you from your templates on a pin router (which is how I'd do it at work).
I regularly cut 2in material in this sort of manner (shapes, that is, not animal outlines), so if you have any more queries, post a response oe email me directly
