Calculating the permitivity of PCBs
Indeed, this is a common way to measure the dielectric constant of materials. I would suggest 'calibrating' your leads, and/or using a 4-wire measurement (provided your LCR meter supports this).
You do need to watch out though: the capacitance of two 10 cm x 10 cm squares on 1.6mm thick FR4 (assuming permittivity of about 4) is about on the order of 0.2 nF, so you need to be carefull with your setup to avoid your measurement being ruined by parasitic capacitance.
Other methods that can be used are making transmission lines (but using a transmission line requires a lot of expertise and precision to get anything useful - you really need to take into account your strip width and profile after etching to get correct results).
A method I've used with some success is by making resonators (usually ring resonators or ground-slot resonators). By matching the measured resonator performance to simulations you can get good results (again, provided you can 'calibrate out' etching effects).