If your family is anything like mine, there is a lot of inherited stuff lingering in cupboards and closets. This may be fine china, silver tea sets, childhood toys, ancient correspondence, favorite mixing bowls... The list goes on.
It's hard to get rid of the things that help inform your identity. And it's hard to keep a lot of moldering or otherwise unlovely stuff that takes up space and never sees the light of day. I'm here to tell you that there are no easy answers - this is an individual journey. There are some strategies, however, for sorting through these things and making decisions you can live with.
First things first: heirlooms you love
First, sort out the stuff you really adore. Don't worry about whether it's valuable, just choose those things that you'd like to use or look at regularly. These go in the keep pile. Then USE THEM. I'm a big believer in using the things you love. I've just inherited the family Wedgewood, which during my childhood was only used on very special occasions. As a result, the set is almost complete. I intend to use it every day. Who cares if it's valuable - I'm not planning to sell it! I'd rather risk breaking or chipping it than leaving it wrapped up for another generation to worry about.
Likewise, if there are family letters you love to look at, put them in an album or otherwise make them accessible (paying attention to using archival materials to keep them safe). Hang the art, put your great-grandmother's wedding dress in a shadow box and hang it in your bedroom! Revel in these inherited treasures.
What not to do? Keep everything, in the name of history! We live in such a disposable society that artifacts from our family's past can seem like rare antiquities. Unless you are very fortunate, they are not. Resist the temptation to keep every item your ancestors touched. Keep the ones that are important to you, the ones you will use, the ones that really do have historical significance, but don't keep it all. I believe in an incremental strategy if this gets difficult. Set aside a few boxes and revisit them over the next several months. As time goes by you will likely recognize that many of the items that seemed so special have lost their magic, and then you'll find it easier to let go of them.
next: the inheritance you just don't want
Inevitably you will inherit stuff you have no use for, and which is not interesting to you. What to do? This stuff is harder, but take heart, you can work your way through it. First, you want to determine whether the items are valuable - either on the market or as historical objects. Letters and other archival information might be of interest to a museum or library, depending on the writer/owner and the contents. My aunt is a historian, and as she's moved into her 80's she's managed to donate lots of books and drawings to various libraries. My favorite? She won a wooden paddle as a camper back in the early 1940's. She just returned it to the camp, still in operation, for their archives!
If you've inherited items that may be valuable, work on getting a real valuation. For some things, this is as easy as checking online. If you use Ebay, make sure to look at the listings that have already sold, to get an idea of what things really go for. If you think your stuff is valuable, especially if there is a lot of it, hire an appraiser. You want someone who is certified, and never hire someone who wants to buy your stuff (conflict of interest). You can use these sites to find someone in your area: International Society of Appraisers, American Society of Appraisers, or Appraisers Association of America. You can use a local auction house or try to sell it yourself online.
I'm sorry to report that many fine antiques have lost value over the last decade or so. Just because they're nice, sadly, does not mean they are worth money. Buck up! Use what you can, sell what you can, release the rest with love and gratitude.
and then, the junk
Realistically there is going to be a lot of stuff that is old but not valuable, or that had sentimental value to the deceased but not to anyone else. It's hard to just dump stuff that was important to your loved one, but be strong! I find that people hate to throw things into the trash - we all want to feel that we are being responsible stewards of the earth, and that we're not tossing out things that still have value. To this I say: recycle what you can, but don't let sentimentality allow you to keep things you really don't want. The landfill is there for a reason.
I've said it before and I'll say it again: keeping stuff in your house that belongs in the trash isn't saving the planet. It's just moving a tiny part of the landfill into your home.