how long can someone in the state of ohio (cuyahoga county) be held for a 3rd degree felony?
That depends on whether you mean held before trial, or upon conviction. In Ohio, a defendant usually must be brought to trial on a felony within 270 days. If a person is being held prior to trial ONLY on that charge, then jail days count 3-to-1, meaning he must be brought to trial within 90 days, absent a waiver of speedy trial.
Upon conviction for an F3, a person can be sentenced to either 1-5 years, or 9-36 months, depending on what he is convicted of.