Required to request Notice of Commencement from GC or required to submit GC a Notice of Furnishing within 21 days in OH?

You can honestly handle this one of three ways. But in order to understand those ways, it's important to understand the purpose of the Notice of Commencement and Notice of Furnishing. A Notice of Commencement (NOC) can be filed by an owner, but it does not have to be filed. If, and only if, an owner files and records a NOC, then, and only then, a subcontractor or supplier (basically anyone who is not directly under contract with the owner) is required to serve a Notice of Furnishing (NOF) if they want to preserve their lien rights. 

With that in mind, the first way to handle the NOF (not recommended at all) is to just ignore everything and not request a NOC or serve a NOF. Some subs and suppliers do this on every project because they have never had an issue in the past. Some do this if they believe the GC to be trustworthy. Some do this if they believe the GC is collectable on a lawsuit, so they don't need the extra pressure of a lien. I personally feel this is the worst way to handle the situation, because payment disputes do arise sometimes, and the amount of effort for the second and third ways laid out below is minimal, while providing immense value to subs and suppliers in the event a project goes south for some reason.

The second way (and the way I think most subs and suppliers should handle this) is, as you mentioned above, to request the NOC from the GC or owner. Ohio has a somewhat convoluted method to accomplish this goal, and the method is laid out in R.C. 1311.04. Basically, in order to request a NOC from the GC or owner, a sub must serve a written request to one of them via certified mail or some other carrier with written evidence of receipt (i.e. FedEx or UPS). Then, pursuant to R.C. 1311.04(D), (E), or (F) (in the case of a subcontractor requesting the notice of commencement from a higher-tiered sub), the higher-tiered party is required to send the NOC to the sub within 10 days. This method is fine, but it does put stress on the NOF timeline. A NOF must be served on the owner within 21 days of commencement of work in order to protect all of a sub's work. If it is served on the 22nd day of work or later, then the NOF only protects work performed in the 21 days prior to service of the NOF. So this method works well, so long as you have an effective staff that can work with strict time constraints and get the request out to the contractor quickly, as well as the NOF served (by certified mail or UPS or FedEx) on the owner quickly, and as long as you have a process or procedure built into every project where someone is tasked with performing the NOC request. In addition, the letter should request any notice of commencement in effect at the time of the letter or any notice of commencement recorded in the future. 

The third way, if the request for a NOC is too time consuming, is to just serve a NOF on every project, regardless of whether a NOC was filed by the owner. The record owner of a property can be found easily using the auditor's page of the county where the work takes place, and service is easy after that. There could be an issue with this method due to the fact that it is possible for a NOF to be served prior to a NOC in some situations, and a court could potentially determine that this sequence is improper (no Ohio court has had the opportunity to review this sequence of events yet). But I would much rather be arguing that position for a subcontractor than be in the position of having no lien rights at all. 

Regardless of how you feel about a GC, a lien is an important tool to have in a sub's possession, because it can help ratchet up the pressure on the owner in the event of payment or project delays. A NOC is recorded on many commercial projects, and service of a NOF is a minimal task if you follow the process outlined in R.C. 1311.04. But your notice of furnishing could be the difference between getting paid or not, so it is absolutely worth spending that extra time and effort on every project and increasing your bid by the $25-50 you will likely spend in costs to prep and serve a notice of furnishing on a project.