First of all, EU citizens are allowed to stay in Czechia indefinitely without any formal registration of temporary residence. (UPDATE: EU citizens in Czechia are required to report their presence to the police, either personally or via their landlord, within 30 days, if they intend to stay longer than 30 days; this, however, is a separate concept, distinct from the temporary residence permit registration.)
Temporary residence is also not needed for renting accomodation. However, such unregistered foreigners won't be able to buy a house / appartment, register a car, get a parking space for residents, and similar.
EU citizens can of course register, if they want to. Since August 2021, they don't need a formal purpose of their stay, such as study or employment (source). That means you don't have to be employed by a Czech company. Actually, you don't have to be employed at all.
However, there are at least two other problems to consider:
Working remotely from another country has tax implications (yes, also within the EU!). I strongly suspect your Greek employer and Greek tax authorities won't approve of it. Please check with them. If you're self-employed (e.g. a freelance software developer working for a Greek company), that's easier, but you'll have to register your business in Czechia and pay taxes there.
You need some form of public health insurance (not the same as travel insurance). Your Greek insurance won't cover much more than emergencies. If you're employed by a Czech company or have your own registered business there, you have to purchase Czech health insurance and cancel your Greek one. It's also possible to purchase health insurance if you're not employed and have no registered business in Czechia, but you'll have to contact the state-owned health insurance company VZP to get more details.