You can use a bed sheet, and there are a few ways to go about it. In almost all the cases, one of the key things to keep in mind is subject separation from the background. You're almost always going to need several feet between the subject and the backdrop/sheet - otherwise it's quite obvious that the subject is just standing in front of a bedsheet.
Option 1) Use a white bed sheet, and light it brightly enough that all the white is blown out. This is good for a "high-key" style which is common in many glam-style portraits. It's very difficult to do without at least two flashes - one for the backdrop and one for the subject - but I guess it's possible if you're creative with your lighting gear. If you have two flashes, use one flash on partial power to light the subject (and a reflector* for fill), and a second flash on high power to light the background, either from the front or the back. With a bed sheet it would probably be pretty easy to light it from the back and blow it all out to white. Check this out for some great theory that you can adapt to your needs, even if you're not using paper you can blow out a white wall or bedsheet just the same way: http://www.zarias.com …torial-part-1-gear-space/
Option 2) Use a white bed sheet, don't light it, and let it fall to gray. in this scenario, you'll light your subject in such a way that virtually no light falls on the backdrop, and use a wide aperture for a shallow depth-of-field so that the wrinkles (etc) in the sheet are out of focus and therefore not objectionable. To accomplish this relatively easily with one light, you'll put your main light to one side of the subject and a reflector* on the other side of the subject. Put the light/reflector as close to the subject as possible without getting them in the frame, both to increase the relative size of the light, and to decrease the necessary intensity of the flash. The backdrop will be very slightly lit from any light that bounces off the walls, ceiling, subject, and reflector, and back onto the backdrop.
Option 3) Black background. This is similar to #2, but use a black bedsheet instead of white. It's tough to accomplish this if you're indoors, as "splashing" and bouncing light will still light the backdrop. If you have a huge room with high ceilings, or if you can set up outdoors at night, it's easier to pull off. A second light source to use as a hair or rim light will really help your subject pop out of the black background, especially if they have dark hair.
* a $2 piece of white foam-core board from the craft store makes a great reflector for things like this