Talking with your neighbors is always the best place to start. Tell them you're worried about their tree and why. Ask them to get the tree inspected by a Certified Arborist and to follow through with whatever recommendations are made.
There are two ways the situation could go after that.
If your neighbors cooperate and agree to get the tree evaluated:
Ask your neighbor for permission to join them when the arborist comes. If the answer is “no,” or if you can’t be there due to scheduling problems, write down any questions you have so your neighbor can get them answered. Ask to see the arborist's report on the tree.
Depending on the arborist's conclusions, you might still be concerned. At this point, you may want to offer to arrange and/or pay for a second opinion. If your neighbors also want a second opinion, offer to split the cost. A friendly gesture like this goes a long way.
Remember: If the situation isn't urgent, be patient. Tree work can be expensive, and everyone should make sure it's needed before getting it done.
If your neighbors ignore you and the tree:
Hopefully this will not be the case, but if it is, you’ll have to take action. Write a letter to your neighbors (or the owners of the property) stating your concerns clearly. Ask them to get their tree evaluated by a Certified Arborist. Tell them that if the tree falls and injures or kills someone, or damages your property in any way, you will hold them responsible. Sign and date the letter, and send it certified (signature required on delivery) to make sure that your neighbors receive it. Send a copy to your insurance company, and keep a copy for yourself. Take some pictures of the tree for your records.
Usually a letter like this gets results. If it doesn't, you may need to consult a lawyer or file a complaint with a government agency. In the City of Atlanta, you can request that the Arborist Division declare a dead tree a "nuisance" and force your neighbors to remove it.