I don't know about you, but I've more than once had the experience of reading a novel set in a particular time period and getting no authentic sense of that period from the text.
I'm not just talking about the Middle Ages here, either (though that too), but about the 1950s, or the 1930s, or the 1890s. And once I start to analyse the reasons for the lack of that sense of authenticity, it comes down to this: the text tells me a lot more about the time in which the author grew up, or currently lives, than it does about the time period it's supposedly set in.
What I mean is that the slang, the cultural references, the attitudes of the characters, and even the characters' names come from, say, the 1970s or 1980s rather than from the setting. To me, this is just as bad as making errors in conveying a sense of place. It reduces the richness of the reading experience.
A lot of people don't care, of course, because they don't know, any more than the author does. But I care, and so do enough other people that I think it's worth getting right - and getting it right is easier now than at any previous time in history.
Unless you confine your stories to the present day and your own familiar locations and culture (which is a perfectly feasible thing to do), you'll need to do some research if you want your story to feel credible. Technology is an obvious area where stories often trip up--I read a book recently set in World War II which mentioned duct tape, an invention of the 1970s. Fashions and cultural references are another. A book set in the 1930s that mentions a "cheerleader skirt," meaning the kind of short skirt that cheerleaders started wearing in the 1970s, fails to convey a sense of its time. You can check these things on Google and Wikipedia very quickly, if you take a moment to think about them.
I'd like to talk specifically about researching names and words, both because they're something that authors often get wrong when setting their stories in an earlier or other time and place, and because they're easy to get right with a couple of simple tools.
Names are powerful, not in the fantasy sense of "if the fairy knows your true name they can control you," but in the sense that they convey a place and time. It's worth putting in some effort with your choice of names, because it's a shorthand way to communicate a sense of authenticity--or, if you get it wrong, inauthenticity.
For example, I read a book set in the 1930s in which two of the characters were named Jason and Samantha. Now, it's possible that there were people with those names in the 1930s, but they would have been very unusual, since both names were rare until they became popular in the 1960s. The author, who is about my age, probably grew up knowing several people with each name, and stuck them into his 1930s story without thinking--and his lack of thought shows.
Likewise, I read a book set in Boston around the time of the American revolution which featured a young woman named Jennifer. Again, this name did exist at the time, but it was almost completely confined to the British county of Cornwall.
Jennifer was the most popular woman's name in the US for nearly a decade and a half, starting in 1970. No doubt when the author was looking for a name for his young woman character, "Jennifer" sprang to mind almost immediately. To his children, though, it won't be a young woman's name, but a name their mother's friends have. And to their children, it'll be an old lady's name. What's more, to someone in 18th-century Boston, it would be rare and exotic.
How do you get this kind of thing right? It's very simple. The website behindthename.com is only one (but, I think, the most useful one) of a number of websites which trace the rising and falling popularity of first names. The best information is for the US, but other countries are also represented.
This brings me to a pet peeve of mine: fantasy novels in which Christianity and Judaism explicitly don't exist, and yet people are called things like Isaac and Maria. It annoys me not for religious reasons, but because it tells me that the author lacks an awareness of culture and history and hasn't thought things through, which instantly lowers my expectations for the book.
Did you know that biblical names weren't even used in England until after the Norman Conquest (1066)? The Saxons who lived there were Christianised, but they still named their children things like Battle, Elf-Counsel, Noble Bright and Wealth-Guard (that's Hilda, Alfred, Albert and Edward to you). The conquerors brought the practice of using biblical names like John and Peter, but also plenty of non-biblical names like William, Henry, Geoffrey and Alice. And then there were the names adopted from Latin and Greek, often saints' names, though not originally Christian: Phoebe, Claire, Nicholas, George, Julia.
It's quite possible to use names that your readers find familiar and can remember without drawing on a cultural heritage that doesn't exist in your setting. And that's a good way to avoid made-up names that are hard to remember (and sometimes unpronouncable without gargling).
Check the spelling of your character names. I once read a book in which the title character's name was spelled wrong four times, despite the fact that it had allegedly been past an editor. Spellcheck won't always catch instances when you've spelled a character's name wrong, especially if your intended spelling is unusual and your typo is the more common spelling, so keep an eye out.
Use of Words
It's easy to spot an author who hasn't thought about or researched their vocabulary. The same book set in the 1930s that I've already referenced twice (for the cheerleader skirt and the characters named Jason and Samantha) also used the phrase "warm fuzzies," which I thought everyone knew arose in the 1970s (clearly, I was wrong). There's an excellent tool to avoid this kind of slip-up. It's called Google Ngram Viewer, and it's at: https://books.google.com/ngrams.
Google has digitised millions of books and other documents, going back as far as 1500, and put them into a searchable database which you can use to find when a word or phrase came into use and the trends in its usage. Using more advanced queries, you can see usage of different parts of speech, like "truck" as a noun versus "truck" as a verb (take a look at "About Ngram Viewer" linked at the bottom of the page for full instructions).
In just a couple of seconds, you can check whether your 1880s steampunk heroine should say "I felt a surge of adrenaline" (she shouldn't; adrenaline wasn't discovered yet).
So if the phrase you want to use wasn't in use yet, what do you substitute? If you want to say "freaked out" (1960s) but your story takes place in 1939, you might find this resource useful:
There's also the Historical Dictionary of American Slang at: http://www.alphadictionary.com/slang/.
Of course, you won't do this kind of research if you don't think about the question in the first place, and this is where the problems start. The past is different. People spoke differently, used different technology, named their children differently, and thought differently about the world. To convey any kind of authentic sense of a past time, you need to be aware that those differences existed, and do the research about what they were.
Reading books and other documents from the time is an excellent way to start to sensitise yourself to possible differences. There are huge digital archives of newspapers and other texts available for free online (maps, too; places change a lot even in a short timespan). And doing this kind of exercise will develop a critical distance between yourself and your own culture, like an observational comedian who asks, "What's that about?" That can only benefit your writing.