Fortunately, both AMD and Intel have been doing a good job at enabling new CPU support under Linux. It's been a couple years since installing any brand new processors and being challenged by kernel panics or other fundamental issues in using new hardware. Both AMD and Intel have been pushing out new product support into the Linux kernel and other areas like code compilers months in advance of the hardware's general availability. Intel has been pushing Haswell Linux support for many months already and on the AMD side they recently have been working on Steamroller / "Bulldozer 3" support.
I'm very happy with Intel Sandy Bridge and Ivy Bridge processors on Linux and have no real issues. On the AMD side, I also like Bulldozer 2 / Vishera a lot along with the Trinity APUs. One word of caution for AMD Linux users is that AMD has been laying off Linux developers and closed their Operating System Research Center but they still do have some Linux developers remaining. There isn't much more to add in this area, although I would certainly avoid VIA hardware even if finding their products at a grand bargain.
Most motherboards and chipsets these days tend to work fairly well under Linux without any serious limitations. Some of the more enthusiast-oriented motherboards may have extra functionality added on that may not be supported under Linux, but in terms of normal use, most modern chipsets/motherboards are in good shape for the Linux desktop. The only other caveats usually come down to hardware sensor support via LM_Sensors/hwmon not always being there right from the start, but only for a small minority of users do they care if they can read their motherboard temperatures and voltages all the time.
In terms of motherboard vendors, I generally find the most Linux-friendly and overall best motherboards out of ASUS. ASUS motherboards are popular with Linux users and there doesn't tend to be many problems. Dealing with ASUS as a company when mentioning Linux also tends to be usually fine and they don't run away or tell you to use Windows. I would avoid Gigabyte as based upon multiple situations, they don't really care about Linux and recommend Microsoft Windows. Tyan, Super Micro, and the other server/workstation-focused motherboard vendors also obviously work quite well on your favorite distribution.
Some motherboard vendors also have BIOS/UEFI issues under Linux, but those tend to be isolated to specific product models rather than being generic to a given vendor. Here are the most common motherboards that were impaired by last year's ASPM power regression problem.