Style and rail doors would be the best solution. Teak or Mahogany wood be the best choice. Solid , as in a slab the size of a door, is difficult to keep from swelling and warping. Style and rail design allows for some swelling of the panels without distortion or warping.
Most cabinet shops are set up to build cabinet doors. There are also custom door builders as well.
You could glue up plywood panels with epoxy, hope they remain stable and the thin veneers don't sand though. It would be best to glue the panels in a press of some kind. The plywood panels are not usually flat enough to glue up without pressure. A flat surface and a stack of 15 sheets or so of 3/4 mdf has worked for me in the past as a poor mans press.
I have built doors using a router and style and rail bits with success, however you need to dowel, or use floating tendons, to tie the rails to the styles.
Another option wood be to just build your rails s4s and use a round over bit to profile the inner edges of the frame and dado the groove for the panel with a router. The panel could be 1/2" teak plywood single or doubled up. Again you would need to use dowels or tendons to tie the frame together. The panels need to float in the frame to allow for movement between the panels and frame.
One of the challenges in building doors is keeping them perfectly flat. all of the solid wood should be machined from rough lumber as surfaced lumber is seldom flat enough to work with. If you have some basic wood working skills and tools, building doors is not rocket science.