Answer: Oregon case law says that holding a check an "unreasonable period of time" is deemed to be acceptance. What is a "reasonable" vs. "unreasonable" period of time? The answer is based on the facts of each case, prior conduct of the parties, and a host of other variables.
However, ORS 90.414 (Acts not constituting waiver of termination of tenancy; delivery of rent refund) gives us some idea what a court would conclude:
(1) If a notice of termination has been given by the landlord or the tenant, the following do not waive the right of the landlord to terminate on the notice and do not reinstate the tenancy:
(a) Except when the notice is a nonpayment of rent termination notice under ORS 90.394, the acceptance of rent if:
(A) The rent is prorated to the termination date specified in the notice; or
(B) The landlord refunds at least the unused balance of the rent prorated for the period beyond the termination date within 10 days after receiving the rent payment.
(b) Except if the termination is for cause under ORS 90.392, 90.398, 90.405, 90.630 or 90.632, the acceptance of rent for a rental period that extends beyond the termination date in the notice, if the landlord refunds at least the unused balance of the rent for the period beyond the termination date within 10 days after the end of the remedy or correction period described in the applicable notice. (Underscore mine.)
Accordingly, under the circumstances you describe, I would conclude that ten days would likely be the outside period for holding a check without either depositing or returning. Although your question doesn'texplain why the rent checks were held, rather than deposited, I'm not sure it makes much difference. Unless there is an ongoing dispute with the resident, say due to an unapproved pet or occupant, good management practice is to either deposit the rent check or return it with an explanation, as quickly as reasonable possible.