What does the lease say?
In no way am I offering you legal advice, and in no way has my comment created an attorney-client relationship. You are not to rely upon my note above in any way, but insted need to sit down with counsel and share all relevant facts before receiving fully-informed legal advice. If you want to be completely sure of your rights, you must sit down with an experienced criminal defense attorney to be fully aware of your rights.
The neighbor that dumped it there is responsible, now to you if you are the one who ultimately disposed of it. That said, yes, check your lease to see what it says as to who is responsible for maintaining the yard. If it is the landlord, then both the landlord and the neighbor are potentially liable to you. You perhaps should have called the police as well - dumping your trash on someone else's property without their permission is likely to be considered criminal.
Nothing contained herein should be considered as legal advice for any specific situation and nothing herein is intended to create a lawyer-client relationship. Every case is very "fact-specific" and persons wishing legal advice on a specific matter should contact me or another attorney for an appointment to review their particular circumstances and to create a lawyer-client relationship. Gregory L. Abbott, Attorney at Law, 6635 North Baltimore, Suite 254, Portland, Oregon 97203. Tel: 503-283-4568; Fax: 503-283-4586; Email: firstname.lastname@example.org. Specializing in Consumer and Small Business Law.
Your neighbor thanks you. Code enforcement would have been my approach and/or your landlord should have responded a little better than he did since ultimately, especially in an apartment complex, keeping the area free of accumulated trash and debris generally falls squarely on the owner/landlord.
A proper response would require a thorough investigation into the history and background of this relationship. The information provided above is just that, information, to be used as you see fit.