I am registering for an LLC after freelancing on the side.
My current clients are writing checks to my name.
Do I just simply cash these checks into my business account going forward?
You need to have your clients now pay the LLC instead of you personally. It should be no big deal for them to switch the payee name.
I'm not an expert on US banking law, but in the two jurisdictions I do know (Canada, UK) you can (unless the check is marked "Not Transferable") simply endorse the cheque by writing:
on the back and then signing the back of the check.
You might want to check with the cashier at your bank - they should know basic laws about checks (but many don't).
(Alternatively you could temporarily pay the checks into your personal account and write equivalent checks from your personal account to the business account - but this is generally not a good idea because it co-mingles business and personal funds and can create problems with disputed transactions.)
Obviously you should now be asking your customers to make any future checks payable directly to "My Company".
Note that you should probably talk to an accountant: in Canada there are rules for converting a "sole proprietorship" into a "corporation" which need to be followed to minimize tax liability.