When you are looking for a way to make very functional FDM parts look closer to being traditionally manufactured parts there are few methods out there. Some of these require manual working of the part, some mechanical, while others require almost no work at all.
One of the major concerns that comes up regularly is the dimensional stability of the parts after the finishing process. When tumbling parts or sanding them you are typically removing small amounts of material from the surfaces, especially the edges, and this can be an issue if not monitored carefully.
Another process that can smooth out parts is to coat the part with either an automotive body filler type substance (ideal for painting parts when dimensions are not critical) or using a two part epoxy type product to smooth and gloss a part over. Both of these methods work, but they are time consuming and will affect the dimensional accuracy of the part(s).