3mm is likely to be too tick to put through a printer. I would imagine that screen printing would be your only option unless you want to look at printing onto a thin vinyl sheet and then bond it to the 3mm thick stuff.
I would be looking at printing onto the vinyl and then laminating it, overall thickness about 3mm. So really I would be printing onto a thinner vinyl and then laminating it. Would have to be durable and pretty strong at sticking to the surface.