I'm sure it can be done. The woven wood shades (that pull up into folds like yours) that we bought for our den could also be bought with a liner. Was going to purchase the same for our bedroom but decided on 2" faux wood blinds instead.
Monica

Yes. We ordered ours from Lowes, and it worked looked great. You can choose blackout, light flitering, etc. They also edge bandings available, wood valances, different fold styles (roman, roll up, etc.).

I just bought some muslin and black cotton material, sewed it together like a pillowcase, inverted it and stiched it to the back of the blinds. I put the white side out and the black side to the back of the blind. I used black thread to weave it into the blind where the other threads were to conceal the additional thread. I cut the fabric where the rings were and resewed them while they were laying on the blind on the floor. A glue gun works well to keep the edges from shifting. It really helps to darken the rooms and insulates a little better. Total cost was about 10.00 for two blinds.

i got mine from budget blinds with the blackout liner, and it works great--i'm sure you could do it yourself.

Funny you should ask this, as I was just thinking yesterday about how dismally I failed at this. I tried doing this myself maybe a year or so ago using some directions I found online. Unfortunately, it didn't work out so well. I used a really thick woven paper sort of material (actually, it was a woven paper window shade that I cut apart). I removed the cords from the woven wood shade, cut slits in woven paper backing where the rings were on the woven wood blind, then used Super 77 spray adhesive to spray the whole back of the window shade and adhere the woven paper backing. It worked great for a while, but over time with raising and lowering the blind, the backing worked loose and started falling down. I took the whole shade down once already and reglued it, but a couple months later, it started coming loose again. It probably would have worked better had I sewed it, but I have next-to-no sewing ability and thought the spray adhesive would work better. Live and learn, I guess. Luckily, the shades (both the woven wood and woven paper) ones were Target cheapies so I won't feel bad replacing them with something else. I'm actually looking at some woven cane paper roller shades from Country Curtains that I think will work perfectly - I just ordered swatches yesterday and am waiting for them to arrive.