I am in no way an expert, but have trained my eye to be able to skim over all the fabrics at Joanns and have an idea of what may or may not be acceptable. None of these are hard and fast rules, but are fairly basic guidelines as to what to look for. I pretty much only do 1860's at this point, so these rules probably won't apply to anything going into the 1870's or later.
Look for strong contrast in printed fabric. In general, go with something that has two very different colors, suhc as orange with blue or red with yellow. Light pink with another medium pink print is probably not a strong enough contrast, although light pink with dark might be okay. The brown one below might have been okay, except that there is hardly any contrast.
Repetitive prints are the very first thing I look for. That is how I can skim so quickly and determine what may or may not be okay. The red one below is not even enough, it looks random. The blue one is actually very good, although I've seen it in person and the dots are a little metallic.
No solid colors for cotton! White is fine, but solid dye in cotton wasn't stable, and would show fading very fast. Chain fabric stores sell fabric that looks pre-faded, so don't use those either. Some dingy colors (like gray-blue) are also not appropriate.
Avoid tonal or teeny-tiny prints, which eliminates just about every single one at Joanns. If it appears solid when you stand back, the print is too small. Anything that is just one continuous solid mass doesn't work with the way fabric was printed back then.
Don't use anything with a black background. Black shows fading very fast, even with print on top. This is one I'm not sure about, but was told.
Be aware of any prints behind the prints, if that makes any sense. There are many wonderful reproduction fabrics that have a print on top of another that is historically accurate, but be aware that you may need to do some research if buying from a chain fabric store.
While floral prints are really pretty, it is veeeerrry hard to find any that are plausible at all. In the 1860's, any flowers were more geometric than realistic, which simply isn't done any more. Realistic is favorited currently, so that is what sells. The one below is a reproduction fabric.
Here is the absolute best place to buy period prints, even if it is expensive: http://www.reproductionfabrics.com/
When shopping online and it gives manufacturer options to limit it down, look for Judie Rothermel and/or Marcus Brothers. Marcus Brothers sells most of their fabric as 'reproduction', but only some of it seems plausible. It is a good place to start, and they do have a few good ones.
Some of the ones that fit all the criteria may not fit your taste criteria. The strong contrast and weird geometric prints sometimes don't match up with our modern color palate, but honestly I've just learned to deal with it and buy the fabric anyway. Wash dresses are supposed to be grubby, get-the-housecleaning-done dresses, and to me it doens't matter all that much if the fabric isn't perfect.
My favorite compromise: go with plaid. It's easier to find a plaid at your local fabric shop that both meets historical accuracy and your favorite colors. WARNING: the homespun plaid Joanns sells can be used, but is of a very cheap quality. Don't pay more than 3-4 dollars a yard, and remember that coupons are your friend. Joanns also takes as many competitor coupons as the amount of items you're purchasing.