As you suggested, a true old fashioned crazy quilt was done as one continuous piece of "made" fabric. Working on an old sheet would give you a stabilizer to help make sure that you keep it nice and flat as you stitch the pieces on.
Another way that I have done crazy is by doing a "mile a minute" ....
http://patchworkpie.blogspot.ca/2008...ute-quilt.html
It's a project that you create over time as you create scraps.
I find it works best to wait til you have a bunch, so you get a good variety of different fabrics.
Then have a sew-fest and put them all together.
Then build up more trimmings and go back at it.
If you don't want defined blocks (with or without sashing), the larger blocks you make the more likely the blocks will just assimilate together to be one.