Their report said blood was found on items including a glucometer for measuring blood sugar.
Faecal contamination was found on toilet equipment such as commodes and raised toilet seats at Ninewells Hospital in Dundee.
They also found "significant levels" of dust on ledges in the special care baby unit, and on two bed frames in the intensive care unit.
The Healthcare Environment Inspectorate (HEI) has detailed nine requirements which NHS Tayside must address as a matter of priority in its report on the unannounced inspection made on March 11-12 this year.
They cover areas from cleanliness to staff dress and breast-milk storage.
Inspectors found there was "poor compliance" with procedures for managing sharp objects such as needles, with bins for them contaminated with blood on the outside. Two were overfilled and the HEI said NHS Tayside must address the problem to cut the risk of injury and infection to patients, staff and visitors.
The frequency of cleaning of the neonatal intensive care unit did not meet national requirements, which the health board has been told to address.
Susan Brimelow, HEI chief inspector, said patients commented positively on the standard of cleanliness on their wards, but added: "This inspection resulted in nine requirements which NHS Tayside must address as a matter of priority."
The inspection team noted not all staff complied with the NHS Scotland dress code.
Six were seen wearing operating theatre clothing in the main concourse area of the hospital. Two doctors with long sleeves were seen in the acute medical unit, and a doctor had untied hair.
NHS Tayside has produced an action plan detailing how it is addressing the nine requirements.
Its chief executive, Lesley McLay said: "We have already addressed the majority of the requirements which have been identified by HEI. The remainder of the issues raised are being actively progressed through our robust action plan, which clearly demonstrates our commitment to further improving our services.
"Our front-line staff work hard every day to ensure person-centred, safe and effective care for all our patients and their families. Clearly, some of the observations noted by the inspectors on the day are disappointing and therefore we are reinforcing policies with staff and providing additional training where necessary."