If you have a gigabit switch connected to non-gigabit router, will the internal network transfer rates be gigabit when the devices are connected to the switch, or will the router also have to have a gigabit port?
Do you have VLAN segmentation? If so - I would assume the 'routing point' for your network is the router, in which case any traffic going between VLANs would traverse the uplink from the GIG switch to the non-gig router. I'm also assuming the speed of your 'non-gig' router is 100mb? Depending on what the network is being used for, and how many users - that 100mb link can be adequate bandwidth or it can be saturated constantly. The only sure way to know would be to check interface statistics during peak transfer periods - on a Cisco Device:
show interface Fa0/x show interface Gi0/x
If the traffic flow is not between VLANs (hosts that are plugged into the switch are all on the same VLANs) - then they would not be using that link to the router as they are all on the same broadcast domain. Assuming all ports negotiated at 1 Gig - you would get 1 Gig speeds.
If this is (as Mike P. classified off topic) a home router/switch question - then I would think the bottleneck would be the ISP speed (unless you've got Gig speeds there).