Do I need a gigabit router to take full advantage of my new gigabit switch, or will my old router do the trick?
As I understand it, if your old router and uplink to the gigaswitch and you aren't using jumbo frames behind the switch, it'll work just fine. You can run into problems if you wanna use a non-standard MTU size behind the switch if the router only supports an MTU size of 1500.
EDIT -- I don't think this statement is true. I have enabled both 4k and 9k jumbo frames in my setup (gigaswitch with JFs and 10/100 router that can't support JFs). Everything works fine with 4k or 9k JFs enabled.
You're going to have really big performance problems if you have a gigabit WAN conection; a gigabit Internet service, but a mesely 10/100 router in front of that. That will cut your Internet surfing performance right down.
But otherwise, what of the communication to the router really needs to go at gigabit speed? Allocating an IP address via DHCP? These things run just fine at 10/100. It's only the local computer to computer transfers that can really run at gigabit speed, and those will go directly from the switch port to the other switch port, without having to make a round-trip through the router.
So short answer -- it'll work just fine.