The mortgage isn't really much of a factor when it comes to liability protection. (But this is really a question to discuss with your attorney. Don't take my word for it.)
An LLC owning the property theoretically creates a "corporate veil" that simply means if you have a claim against one of your properties (a personal injury or wrongful death claim being the worst case scenario that most landlords are looking to protect themselves from), the claimant can not go after your personal assets, only the assets owned by the LLC.
But there's more to it that just placing the deed in the name of an LLC. You also need to keep separate accounts and bookkeeping for each LLC, and avoid any commingling of funds. Using the money in your LLC's bank account to buy yourself a pair of sneakers or pay your personal electric bill can make your corporate veil easier for a plaintiff's attorney to "pierce". (An attorney could argue that your LLC is total BS, as evidenced by the fact you used it to buy yourself Taylor Swift tickets, and a judge or jury might agree).
Another crucial component of liability protection is your insurance coverage. Many investors make the mistake of changing the deed over to an LLC, but leaving the insurance policies in their personal name. This can create the risk of claims being denied on the basis of you (personally) not having an insurable interest in the property, since it is not owned by you; It is owned by a corporate entity. Much like you cannot take out an insurance policy on my house (and expect to be paid when there is a claim), because you don't own my house.
Also note that if you ask an attorney, a CPA, and a mortgage broker how to structure your LLCs, you're likely to get three different answers. There are liability issues to consider (so the attorney will tell you one thing), tax implications (so the CPA might tell you another), and financing limitations (you can't get a conventional mortgage in the name of an LLC, and the due on sale clause noted by another poster) to consider. So it's important to understand the nuances and figure out what works for you, because there is no one size fits all solution.