You can accept partial payments without prejudice to you unless they say on the reverse "full accord and satisfaction" which would indicate that cashing the check satisfies the payment obligation in full.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
You may want to consider looking at your agreements with your customers to provide that you can charge interest on late or outstanding accounts and collect attorney fees if you have to sue them to collect the amount owed. You can also add a provision that if they write "Paid In Full" or "Accord and Satisfaction" on the check that you may cash the check but it does not change their obligation to you unless you consent in writing.
You should look at cleaning up your current accounts and employing a mechanism to put you in a better position if your customers fail to uphold their end of the bargain.
The response is based on the information provided. Absent an express agreement, this communication does not constitute legal advice or create an attorney-client relationship.
When a customer forwards you a check, please be sure the following language is not included "paid in full," or something to that effect. In the future, if a customer is to submit partial payment, you should consider entering into an agreement expressly establishing the said terms.
The information is for general information purposes only. Nothing from this comment should be taken as legal advice for any individual case or situation. This information does not create and does not constitute an attorney-client relationship or a prospective attorney relationship.