I had various faults with my first three, before I got a good one. Some problems people have experienced are due to hardware, some firmware, and some software. These are my observations about my own experiences and common problems I have read about:
Hardware/firmware issues
Random graphics freezes with 2600XT mac pro usually requiring hard reboot. This was an extremely common problem among 2600XT owners, and my second one did this. I don't know if the 2600XT update fixed it as I had long since exchanged mine for an 8800GT, but I know that some people were still reporting freezing after updating (as well as more fan noise and power draw).
Reboot or freeze when waking from sleep. This was another very common problem, and also unpleasantly random, but it sounds like it has been fixed with the recent firmware update. I never use sleep so it didn't bother me anyway.
Optical drive issues, I've had 3 optiarcs and 1 pioneer in my mac pros. The pioneer let me rip one CD and then died, but I think that was just isolated bad luck, or a faulty logic board as other weird things were happening too. My current optiarc works fine with all purchased disks (although it's sometimes very loud) but it can only recognise about two thirds of the many backup disks that I burnt in the past using my powermac G4 with a pioneer superdrive. I saw a post where someone else had the same issue and apparently when they reported it Apple told them that they don't guarantee the optical drive will read disks burnt using any other drive. I wouldn't mind so much if I had used cheap disks at top speed with no verification but these were quality verbatim dvd-r that were verified after burning using toast and superdrive and I never had one coaster. The pioneer can still read them, why is it unreasonable to expect a brand new drive to read them? I think Apple are just using the cheapest optical drives that the manufacturers have left over. This is the only hardware problem I have with my current mac pro and since I can fix it cheaply by buying another drive I don't intend to waste more time returning it, and risk getting another machine with a worse fault. It took them 13 days to replace my third one, despite their promises.
Hard disk performance; the stock drive is very slow. It amazes me that Apple reduce the performance of a "pro" machine by using such drives to save a handful of dollars on a machine that is this expensive. I almost wonder if the disk manufacturers are paying Apple to take their oldest junk off their hands, because it's cheaper than landfill tax! Some people have found the seagates to be very noisey, I have had 3 WDs and 1 seagate, and the seagate was only a little bit louder than the WDs.
Bluetooth issues. I don't use bluetooth so I can't comment but I have seen other owners complaining about problems. I don't know if it's hardware or software related, maybe it's just some Leopard bug, or perhaps there's just too much damn metal around the antenna!
Noise. This has not been a problem on any of my four mac pros, but some people have had knocking fans etc. Mine were all similar volume to each other, certainly not silent, if you are in a quiet environment you can easily hear that it is switched on but it is an ambient sound that you quickly tune out, a bit like the quiet sound of water flowing through a radiator in the room - you just forget it's there. The only thing that is noisy is the optical drive when spinning at high speed.
Overheating. My first mac pro had a power supply where both its sensors read twice as hot as most mac pros. I have only seen one other person report that issue. My other mac pros have all been fine. Some people have had concerns about RAM temperature, particularly with extra RAM fitted. I'm using stock 2GB RAM and the temps have been fine. A lot of people manually increase the RAM fan speed when they fit extra RAM, and it does seem odd that Apple did not program the fan speed to ramp up automatically when RAM temperatures begin to get frightening.
Software issues
I have seen a few weird issues with Leopard, even on fresh installs to a zero'd disk. The automatic software update is very flakey. Sometimes I have done a clean install, then tried to apply the updates, only to have it get stuck forever with no desktop and the white spinning thing when it should be closing to reboot. It is definitely a good idea to get 10.5.2 combo from the downloads website rather than the autoupdate. Incidentally it is a shame it takes so long for Apple to update the pre installed software and the restore disks in the mac pro box. Many weeks after 10.5.2 was released they still send "Just In Time" manufactured BTO computers out the factory with 10.5.1. Since 10.5.2 is over 300mb it takes a long time to download if you don't have much bandwidth (another good reason to get the combo updater as you can burn it if you plan to do more fresh installs in the near future).
I see occasional GUI bugs, such as icons becoming invisible in dock stacks. I'm sure Apple are gradually ironing out stuff like that, but it is disappointing since Leopard is hardly new anymore, and you would expect a clean install (after updates) to be almost flawless before you add any third party software.
A couple of months ago there seemed to be big problems with Leopard randomly saying "you are not connected to the internet" even when you clearly are and other machines on the same router are connecting fine. This issue seems to have gone now so it was either an early leopard bug or a hardware issue with the first macpro I had.
Software installation duration. I don't know if this is due to the software disks being highly compressed, or the optical drive being slow, or the hard drive being slow, but installing Logic studio (about 6 and a half dvds) takes me over 5 hours. I have seen other people reporting this too, so it's not just me. I have done this about 6 times in two months while testing and returning faulty mac pros and it has been a colossal waste of time (combined with fresh installs of Leopard and about 1.5GB of software updates to download just for Leopard/bundled apps and Logic Studio).
Advice
If you have an apple store close by and you don't need BTO options then buy it in the store, not online. Then if there is a problem you just take it back for replacement or refund. If you get it online and there is a problem, say you need to swap one little component, the local Apple store will probably tell you they can't swap that one component nor swap the machine (since you didn't buy it from their store and they consider the whole machine to be one product) so you will have to return the whole damn machine by courier.
Within 14 days you should run the hardware test and also try using every part of it that you are ever likely to want to use (e.g. burn a disk, try all the graphics/firewire/usb ports, check the temperatures etc). That way if you find something wrong you won't need to waste time while it sits in a repair shop, you can just demand an immediate replacement or refund within 14 days.
If you do notice something weird but it is not crippling you then try to determine if it is likely to be hardware, firmware or software at fault. If it's software it can probably be fixed either by update or OS reinstall, if it's firmware it might be fixed but you may have a long wait and apple won't acknowledge there is a problem until the fix is ready, if it's hardware then send it back. The recent 2600XT freeze issue (and I would imagine the reboot on wake issue) put owners in a real quandry. I suspected these issues could be fixed with firmware, but I wasn't prepared to gamble £1750 on it, nor to put up with weeks or months of my machine freezing up every hour or so and me losing my work as I was forced to hard reboot. Apple did not say "don't worry guys, we acknowledge that there is a problem, we think we can fix it, just hang on a while", they said absolutely nothing until firmware updates appeared 2-3 months later. Buyers in this period did not know if they would be left with hardware that would never work as it should (apple have sometimes left owners screwed like this in the past with flawed hardware) so a lot of them bailed out before it became too late to bail.
Early 2008 mac pros all look the same but it seems they are all unique and have little quirks of their own. I hope you are lucky enough to get one first time whose quirks are acceptable to you. I am enjoying mine now, and with a better optical drive it will be fantastic.