The guys down at the natural history and fine art museums are wizards at doing that kind of stuff. Their favorite material is either pre-laminated or uncoated MDF. If needed prime and paint the MDF, then mount the photos with aggressive slightly acidic adhesive. Leave plenty of image bleed all round, trim carefully with an Xacto knife. Archival be damned, better to have those prints stay in place for a few years that to peel off in archival condition!
The only real downside with MDF is that the cut, cross sectional sides with the sawdust exposed is exceedingly porous and will soak up a lot of primer and paint, maybe 20 times more than the paper coated surface! It's very difficult to make a face-to-side joint with MDF and have it look good through the paint. So plan to work at it. If you can build your MDF boxes with 45 degree miter cuts on the table saw, go for it but that ain't easy. But nothing works better for that kind of app.
A slightly easier alternative for an in-depth effect is to mount the prints on aluminum panels or Gator, then bring dowels out from the wall for support.