Buy a sheet of extruded foam insulation board 3/4" thick. (not white styrofoam beadboard) If you cut it carefully, it's a press-fit in the jamb. Plastic sheet is OK to keep bugs out, but no security or privacy. If security is the #1 concern, you'll have to plywood over the opening.
Casey

Great idea! Thank you, sombreuil mongrel.

What I do for security, i.e. ground floor is make a plywood cover that fits into the jamb (or lies against exterior trim, whichever is better for your project and weathertightness), but is securely held into the opening buy using long carriage bolts that run from the plywood and extend far enough into the room that they go through a 2 x 4 that lies aginst the inside of the window casing parallel to floor. (Tall windows need two 2 x 4s, smaller windows can get by with only one.) This allows you to secure the plywood without making any nail holes on the trim. Plus if you have same-sized windows, you can just reuse the plywood for other windows. If you start on your larger windows and complete them first, you can just keep cutting down the plywood and 2 x 4s as you go on to smaller windows. Or if you have many windows, you could keep one of each size to use as emergency repairs.

What this contraption is (from outside) 1 X 4 lying flat against plywood; washer and bolt running from 1X4, through plywood, exposed within window opening, then through 2X4 with large washers and nut on end. I have simple D-handles mounted on inside of plywood to make it easier for helper to grasp it while it's being installed and snugged up. You can weatherstrip this with felt or foam strips if it was cold weather. The plywood and lack of light is a great motivator to keep the project moving along as it is U.G.L.Y!

If you go the plastic route then I suggest you use greenhouse batten tape (a fairly thick white plastic band that's used to hold down plastic covers on greenhouses). This can be stapled up with plastic and only makes small easy to patch holes on trim. It's designed for holding sheet plastic taut and works very well. If there's any chance you'll have the windows out for more than a few weeks, I'd also suggest using te more expensive UV-stabilized plastic films designed as greenhouse covers. Both of the products could be purchased at greenhouse suppliers.

I have also used something called a WindoTherm panel in a custom-made, non-marking set-up installed from the inside. One winter this was my only covering on a bathroom window and it did just fine, even though it was only two layers of plastic film. Cats are hard on Windotherms however.

HTH,
L

liriodendron - thanks for your description of the plywood cover. Last year when I was working on a couple back windows and had to prepare to leave the house for a week, I devised something like that but fortunately didn't have to use it.

I wish I had known about batten tape last year. I used duct tape and still have glue residue on the house.

I typically fill the frame with a piece of 2 inch foam board cut to a tight jam fit in the opening.

If there are no storms some 1/2 inch plywood sized to fit the rabbets for wooden storms, or to completely cover the opening.

I use smaller diameter square drive screws ('trim head') and just count on filing holes and painting the wood after removal of the plywood and re-installation of the repaired sashes.