Gigabit Ethernet has now become quite commonplace, especially since the prices of switches are dropping so rapidly.
Cabling issues. There are some minor problems with various types of cabling. It requires at least Cat-5 cabling to be used, although for optimal performance Cat-5e or Cat-6 (and maybe the future Cat-7) should be used. Gigabit ethernet is also quite a bit pickier about the cable being crimped right, since it modulates the medium so much faster... so cable that worked well with 10Mbit or Fast Ethernet may not work quite so well with gigabit.
Compatibility. For the most part, backwards compatibility is maintained. However, I have run into several issues, especially with cheap network cards, mostly dealing with doing the proper negotiation for a 100Mbit card on a 1000Mbit network. However, I have run into a few problems with a Cisco Catalyst, where the port was wired for 100Mbit and it would not speak to a 1000Mbit card (so I put a cheap 100Mbit switch between them).
Switches. I am mostly concerned with dumb, layer 2 switches. So far, the ones that I have seen work the best are any of the Netgear models (especially the pretty blue ones). They have recently released their fanless models, such as the GS105, a rather cheap ($80 or so) 5 port model.
NICs. Many systems are now getting gigabit ethernet thanks to the Broadcom 570x series of chips, since they are being integrated into so many systems and laptops nowadays, instead of the aging Realtek 100Mbit sets of yore. These chipsets perform well, with good bandwidth and latency. However, if you are installing a new system, you can do worse than to use the mighty Intel gigabit ethernet cards. They tend to run a bit more cost-wise, but make up for it in raw performance and stability. The server (64-bit PCI-X) models are particularly nice. I have had a lot of issues with Netgear NICs though, even the PCI-X models (systems not always POSTing, general instability with Windows, improper negotiation of speed).
Another concern of many people switching to gigabit ethernet is the infamous jumbo frame. The standard Ethernet frame is 1,500 bytes long, but many gigabit NICs support the so-called jumbo frame, of usually 15,000 bytes. Supposedly, this can boost bandwidth. However, in order to actually utilize the bandwidth, you would need to be relatively close to the end point you want to talk to (such as a server). Once you have to talk down to non-jumbo frame devices (routers and such), there is no point in using jumbo frames, since they won't understand frames that large. And a lot of times the bandwidth utilization is a problem related to a bottleneck in the system itself: slow hard drive, PCI bus getting flooded, etc. And considering Windows has a default TCP sliding window size of only around 8000, jumbo frames won't matter much.